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LIDS- P 859
Research Supported By:
ERDA-E contract (49-18)-2087
A Schur Method for Solving Algebraic Riccati Equations*
Laub, A.J.
A SCHUR METHOD FOR SOLVING ALGEBRAIC RICCATI EQUATIONS*
by
Alan J. Laub**
ABSTRACT
In this paper a new algorithm for solving algebraic Riccati equations
(both continuous-time and discrete-time versions)
is presented.
The
method studied is a variant of the classical eigenvector approach and
uses instead an appropriate set of Schur vectors thereby gaining substantial numerical advantages.
number of numerical issues.
Considerable discussion is devoted to a
The method is apparently quite numerically
stable and performs reliably on systems with dense matrices up to order
100 or so, storage being the main limiting factor.
*This research was supported by the U.S. Energy Research and Development
Agency under contract ERDA-E(49-18)-2087.
**Laboratory for Information and Decision Systems, Rm. 35-331, Massachusetts
Institute of Technology, Cambridge, MA 02139; ph.:
(617)-253-2125.
-2-
1.
Introduction
In this paper a new algorithm for solving algebraic Riccati equations
(both continuous-time and discrete-time versions) is presented.
These
equations play fundamental roles in the analysis, synthesis, and design
of linear-quadratic-Gaussian control and estimation systems as well as
in many other branches of applied mathematics.
It is not the purpose of
this paper to survey the extensive literature available for these equations
but, rather, we refer the reader to, for example,
[5] for references.
[1],
[2], [31,
[4], and
Nor is it our intention to investigate any but the
unique (under suitable hypotheses) symmetric, nonnegative definite solution of an algebraic Riccati equation even though the algorithm to be
presented does also have the potential to produce other solutions.
For
further reference to the "geometry" of the Riccati equation we refer to
[31,
16], and [7].
The method studied here is a variant of the classical eigenvector
approach to Riccati equations, the essentials of which date back to at
least von Escherich in 1898 18].
The approach has also found its way
into the control literature in papers by, for example, MacFarlane
Potter 110],
and Vaughn 111].
[9],
Its use in that literature is often
associated with the name of Potter.
However, the use of eigenvectors
is often highly unsatisfactory from a numerical point of view and the
present method uses the so-called and much more numerically attractive
Schur vectors to get a basis for a certain subspace of interest in the
problem.
Other authors such as Fath [121
and Willems
[31,
to name two, have
also noted that any basis of the subspace would suffice but the specific
-3-
use of Schur vectors was inhibited by a not-entirely-straightforward
problem of ordering triangular canonical forms - a problem which. is
discussed briefly in the sequel.
The paper by Fath is very much in
the spirit of the work presented here and is one of the very few in the
literature which seriously addresses numerical issues.
One of the best summaries of the eigenvector approach to solving
algebraic Riccati equations is the work of Martensson [13].
This work
extends [10] to the case of "multiple closed-loop eigenvalues."
It
will be shown in the sequel how the present approach recovers all the
theoretical results of [10]
and
[131 while providing significant numeri-
cal advantages.
Most numerical comparisons of Riccati algorithms tend to favor the
standard eigenvector approach - its numerical difficulties notwithstandingover other approaches such as Newton's method
integrating a Riccati differential equation.
[7],
[15], and [16].
[14] or methods based on
Typical of such comparisons are
It will be demonstrated in this paper that if you previ-
ously liked the eigenvector approach, you must prefer, almost by definition, the
Schur vector approach.
This statement, while somewhat simplistic, is based
on the fact that a Schur vector approach provides a substantially more efficient,
useful, and reliable technique for numerically solving algebraic Riccati equations.
The method is intended primarily for the solution of dense, moderate-
sized equations
(say, order < 100) rather than large, sparse equations.
While
the algorithm in its present state offers much scope for improvement, it still
represents a substantial improvement over current direct methods for solving
algebraic Riccati equations.
-4-
Briefly, the rest of the paper is organized as follows.
is concluded with some notation and linear algebra review.
This section
In Sections
2 and 3 the continuous-time and discrete-time Riccati equations, respectively, are treated.
In Section 4 numerical issues such as algorithm
implementation, balancing, scaling, operation counts, timing, storage,
stability, and conditioning are considered.
In Section 5 we emphasize
the advantages of the Schur vector approach and make some further general
remarks.
Six examples are given in Section 6 and some concluding remarks
are made in Section 7.
1.1
Notation
Throughout the paper A e
efficients in a field
IR
IF.
or the complex numbers
IF
will denote an mxn matrix with co-
The field will usually be the real numbers
C.
The notations AT and AH will denote
transpose and conjugate transpose, respectively, while A
(A )
= (A
)
.
The notation A
-T
will denote
will denote the Moore-Penrose pseudo-
inverse of the matrix A.
For A e
will be denoted by C(A).
When a matrix A e
nxn its spectrum (set of n eigenvalues)
IR2nx2n is partitioned into
four nxn blocks as
A=
112
A128
21
A22/
A
we shall frequently refer to the individual blocks A..i
discussion.
without further
-5-
1.2
Linear Algebra Review
Definition 1:
A e
Definition 2:
A e
Let J =
IR
= A- 1
is orthogonal if A
nxn
H
-l
x is unitary if A = A
IR2 nx2 n where I denotes the ni- order identity matrix.
e
0
I
Note that J
T
=
-l
= -J.
Definition 3: A e
IR
is Hamiltonian if J
Definition 4: A e
IR
is symplectic if J-1 A J = A - 1
Hamiltonian
A TJ = -A.
and symplectic matrices are obviously closely related.
For
a discussion of this relationship and a review of "symplectic algebra"
see
[17],
algebra.
[18].
We will use the following two theorems from symplectic
Their proofs (see [18]) are trivial (and hence will be omitted).
Theorem 1:
1.
Let A e
IR
be Hamiltonian.
implies -X e C(A) with the same multiplicity. 2.
symplectic. Then X e CG(A)
implies
X
Then X e CT(A)
Let A e
IR2 n x 2n, be
e aC(A) with the same multiplicity.
There is a relationship between the right and left eigenvectors of
these symplectically associated eigenvalues.
Theorem 2:
U e
Let A e
IR
R 2 n x 2 n be symplectic.
See [18]
be Hamiltonian
for details.
(or symplectic).
Then U-1 AU is Hamiltonian
Let
(or symplectic).
Finally, we need two theorems from classical similarity theory which
form the theoretical cornerstone of modern numerical linear algebra.
See [19], for example, for a textbook treatment.
-6-
Theorem 3 (Schur canonical form):
Xl,...,X .
Let A e
Rxn have eigenvalues
Then there exists a unitary similarity transformation U such
that U AU is upper triangular with diagonal elements Al,...
k
n
in that
order.
In fact, it is possible to work only over
IR by reducing to quasi-
upper-triangular form with 2x2 blocks on the (block) diagonal corresponding
to complex conjugate eigenvalues and lxl blocks corresponding to the real
eigenvalues.
We refer to this canonical form as the real Schur form
(RSF) or the Murnaghan-Wintner
Theorem 4
(RSF): Let A e
canonical form.
[.20]
nR
nxn
.
Then there exists an orthogonal
T
similarity transformation U such that U AU is quasi-upper-triangular.
Moreover, U can be chosen so that the 2x2 and lxl diagonal blocks appear
in any desired order.
T
If in Theorem 4 we partition U AU into
11
S12
kR
O<k<n, we shall refer to the first k vectors of U as
corresponding to C(S ll) C G(A).
where Sll
e
kxk
the Schur vectors
The Schur vectors corresponding to the
eigenvalues of Sll span the eigenspace corresponding to those eigenvalues
even when some of the eigenvalues are multiple (see [21]).
use this property heavily in the sequel.
We shall
-7-
2.
The Continuous-Time Algebraic Riccati Equation
In this section we shall present a method for using a certain set of
Schur vectors to solve
(for X) the continuous-time algebraic Riccati
equation
F X + XF - XGX + H = O .
All matrices are in
]R
n
(1)
and G = GT
> 0, H = H
> 0.
It is assumed that (F,B) is a stabilizable pair [1] where B is a
full-rank factorization (FRF) of G(i.e., BB
and
(C,F) is a detectable pair
and rank(C) = rank(H)).
[11
T
= G and rank(B) = rank(G))
where C is a FRF of H (i.e.,
Under these assumptions,
a unique nonnegative definite solution
1J1.
C C = H
(1) is known to have
There are, of course,
other solutions to (1) but for the algorithm presented here the emphasis
will be on computing the nonnegative definite one.
Now consider the Hamiltonian
Z
<
(
F
-GA
e
-H
matrix
2nx2n
R2 nx 2 n
(2)
-F/
Our assumptions guarantee that Z has no pure imaginary eigenvalues.
Thus by Theorem 4 we can find an orthogonal transformation U e
R 2 nx 2 n
which puts Z in RSF:
S1:2
12
=ll
U ZU = S =
\°
where S.. e
1j
R nx
(3)
S22/
It is possible to arrange, moreover, that the real
parts of the spectrum of Sll are negative while the real parts of the
spectrum of S22 are positive.
nxn blocks:
U is conformably partitioned into four
-8-
U
11
U12
U21
U2 2
We then have the following theorem.
Theorem 5:
1.
Ull is invertible and X=U1U
2.
Proof:
With respect to the notation and assumptions above:
(Sll)
-1
1 solves
T
(1) with X=X >0.
= c(F-GX) = the "closed-loop" spectrum.
A direct proof of this theorem may be found in [22]
omitted here.
but will be
An alternate proof is suggested in Remark 1 below.
Remark 1: As an alternative to the direct proofs provided in [22] one
could simply appeal to the proofs given for the eigenvector approach and note
that the Schur vectors are related to the eigenvectors by a nonsingular
transformation.
Specifically, with Z, U, and S as above, let V
IR 2
2n
put Z in real Jordan form
1
(5)
(ZV
-A=
(gnx2n denotes the set of 2n x 2n matrices of rank 2n, i.e., invertible)
2n
where -A is the real Jordan form of the eigenvalues of Z with negative real
parts.
-A.
Furthermore, let T
epxn transform
n
S
11
to the real Jordan form
Then
Z
21
(21-A)
(6)
and
Z (Ull) =(Ul
21W
v11e
tU21
We thus have
2
21:3;t1
11··~
-9-
1~T (-A)
.(7)
2J
Since eigenvectors are unique up to nonzero scalar multiple, comparing (6) and
(7) we must have
(1
T=
(V1)DT
where D is
V
-1
Vl1
21 11
solves
-1
21 11
diagonal and invertible,
(1), U
-1
Ull must also solve
21 11
-1
21
Thus
-1 -1
11
11
=Vll
-1
and since
(1) since
-1
21 11'
Further discussion of Theorem 5 and computational considerations are
deferred until Section 4.
-10-
3.
The Discrete-Time Algebraic Riccati Equation
In this section we shall present an analogous method using certain
Schur vectors to solve the discrete-time algebraic Riccati equation
TT
T
-1 T
- X - FXG(G 2 + GlXG)
GXF + H = 0.
FTF
Here F,-KHX £RE
Also, m<n.
nxn
RR
' G1 £
nxm
G2
IR
mxm
, and H=H
(8)*
T
T
>0, G2=G >0.
The details of the method for this equation are sufficiently
different from the continuous-time case that we shall explicitly present
most of them.
It is assumed that (F,G1 ) is a stabilizable pair and that (C,F)
is
a detectable pair where C is a FRF of H (i.e., C C=H and rank (C)= rank (H)).
We also assume that F is invertible - a common assumption on the openloop dynamics of a discrete-time system [23].
when F is
The details for the case
singular or ill-conditioned with respect to inversion can be found
in [24].
Under the above assumptions
(8) is known to have a unique nonnegative
definite solution [25] and the method proposed below will be directed towards
finding that solution.
-1 T
Setting G= G G
G we consider this time the symplectic matrix
12 1
(F+GF TH
-F
H
(9)
-GFT
F
Our assumptions guarantee that Z has no eigenvalues on the unit circle.
Theorem 4 we can find an orthogonal transformation U E IR
2nx2n
which puts
Z in RSF:
*Note that an alternate equivalent form of
T
FT(X
T-1 - G)
T-=
+GG
1 2 G1
F
F-X+H=0
By
(8) when X is invertible is:
UTZU = S
0
where Sij £ R
nxn
(10)
12
11
S22
.
It is possible to arrange, moreover, that the spectrum of Sll lies
inside the unit circle while the spectrum of S22 lies outside the unit
circle.
Again U is partitioned conformably.
We then have the following
theorem.
Theorem 6:
With respect to the notation and assumptions above:
1.
-1
Ull is invertible and X=U21Ull solves
2.
C(Sll)= O(F - G (G2 +G XG )
T
= Q(F - GF
-T
= a(F - G(X
-1
T
(8) with X=X >0.
-1 T
G1XF)
(X-H))
+ G)
-1
F) when X is invertible
= the "closed-loop" spectrum.
Proof:
As for Theorem 5 we shall omit a direct proof of this theorem and refer
the interested reader instead to [22] or [24].
Again an alternate proof is
possible as in Remark 1.
We now turn to some general numerical considerations regarding the
Schur vector approach.
-12-
4.
Numerical Considerations
There are two steps to the Schur vector approach.
The first is re-
duction of a 2nx 2n matrix to an ordered real Schur form; the second is
th
the solution of an n - order linear matrix equation.
We shall discuss
these in the context of the continuous-time case noting differences for
the discrete-time case where appropriate.
4.1
Algorithm Implementation
It is well-known
(see [21], for example) that the double Francis
QR algorithm applied to a real general matrix does not guarantee any
special order for the eigenvalues on the diagonal of the Schur form.
However, it is also known how the real Schur form can be arbitrarily reordered via orthogonal similarities; see [21] for details.
further orthogonal similarities required to ensure that
Thus any
C(Sll) in
(3) lies
in the left-half complex plane can be combined with the U initially
used to get a RSF to get a final orthogonal matrix which effects the desired ordered RSF.
Stewart has recently published FORTRAN subroutines
for calculating
and ordering the RSF of a real upper Hessenberg matrix 126].
The lxl or
2x 2 blocks are ordered so that the eigenvalues appear in descending order
of magnitude along the diagonal. Stewart's
software
(HQR3) may
thus
be used directly if one is willing to first apply to the Z of (2) an
appropriate bilinear transformation which maps the left-half-plane to
the exterior of the unit circle.
Since the transformed Z is an analytic
function of Z, the U that reduces it to an ordered RSF - with half the
eigenvalues outside the unit circle - is the desired U from which the
solution of (1) may be constructed.
Alternatively, Stewart's software
can be modified to directly reorder a RSF by algebraic sign.
In the discrete-time case, HQR3 can be used directly by working
with
-1
Z
=
/F-1
-1
FG-1
T HFG
(11)
The U which puts G(Sll) outside the unit circle is thus the same U which
puts the upper left nxn block of the RSF of Z inside the unit circle.
In summary then, to use HQR3 we would recommend using the following
sequence of subroutines
(or their equivalents):
BALANC
to balance a real general matrix
ORTHES
to reduce the balanced matrix to upper Hessenberg
form using orthogonal transformations
ORTRAN
to accumulate the transformations from the Hessenberg
reduction
HQR3
to determine an ordered RSF from the Hessenberg matrix
BALBAK
to backtransform the orthogonal matrix to a nonsingular matrix corresponding to the original matrix.
The subroutines BALANC, ORTHES, ORTRAN, BALBAK are all available in
EISPACK [27].
th
The second step to be implemented is the solution of an n - order
linear matrix equation
XU
11
U
21
-1
to find X = U21Ull.
For this step we would recommend a good linear
equation solver such as DECOMP and SOLVE available in 128] or the appropriate routines available in LINPACK 1291.
A routine such
as DECOMP computes the LU-factorization of Ull and SOLVE performs the
forward and backward substitutions.
A good estimate of the condition
number of Ull with respect to inversion is available with good linear
equation software and this estimate should be inspected.
A badly con-
ditioned Ull usually results from a "badly conditioned Riccati equation".
This matter will be discussed further in Section 4.4.
While we have no
analytical proof at this time, we have observed empirically that a condition number estimate on the order of 10t
for Ull usually results in
a loss of about t digits of accuracy in X.
One final note on implementation.
Since X is symmetric it is usually
more convenient, with standard linear equation software, to solve the
equation
T
T
UllX = U2
11
21
to find X =U
4.2
-TT
-1
Ul =U U21
11 21
21 11'
Balancing and Scaling
Note that the use of balancing in the above implementation results
in a nonsingular (but not necessarily orthogonal) matrix which reduces
Z to RSF.
More specifically, suppose P is a permutation matrix and D is
a diagonalmatrix such that PD balances Z, i.e.,
D
PZPD = Zb
where Zb is the balanced matrix; see [303 for details.
an orthogonal matrix U which reduces Zb to ordered RSF:
UTZbU
= S
We then find
-15-
Then PDU (produced by BALBAK) is clearly a nonsingular matrix which
reduces Z to ordered RSF.
The first n columns of PDU span the eigenspace
corresponding to eigenvalues of Z with negative real parts and that is
the only property we require of the transformation.
For simplicity in
the sequel, we shall speak of the transformation reducing Z to RSF as
simply an orthogonal matrix U with the understanding that the more computationally attractive transformation is of the form PDU.
An alternative approach to direct balancing of Z is to attempt
some sort of scaling in the problem which generates the Riccati equation.
To illustrate,
consider
the linear optimal control problem of finding
a feedback controller u(t) = Kx(t) which minimizes the performance index
J(u)
=
fT
Ix (t)Hx(t) + u
T
(t)Ru(t)Jdt
with plant constraint dynamics given by
x t) = Fx(t) + Bu(t)
We assume H = H
;
> 0, R = RT
x(.)
> 0 and
where C C = H and rank(C) = rank(H).
= x
0
(F,B) controllable, (F,C) observable
Then the optimal solution is well-
known to be
u(t) = -R lBTXx
where X solves the Riccati equation
T
F X + XF-
XBR
-l T
B X + H
0
Now suppose we change coordinates via a nonsingular transformation
x(t).
Tw(t).
minimize
Then in terms of the new state w our problem is to
-16-
Of
[w(t)(TTHT)w(t) + uT(t)Ru(t)ldt
subject to
O(t) = (T
FT)w(t) +
(T
B)u(t)
.
The Hamiltonian matrix Z for this transformed system is now given
by
/-1
-l T
(
w
T HT
HT
-1
1 T -T
-T-1BR BT-T
TTFTT-T
-T F T
and the associated solution X of the transformed Riccati equation is
w
related to the original X by X = T
X T
.
One interpretation of T then
is as a scaling transformation, a diagonal matrix, for example, in an
attempt to "balance" the elements of Z .
Applying such a procedure, even
w
in an ad hoc way, is frequently very useful from a computational point
of view.
Another way to look at the above procedure is that Zw
T) , i.e.,
is symplectically similar to Z via the transformation
T
Z
=
-T
It is well-known that Z
Z
T·
is again Hamiltonian (or symplectic in the dis-
crete-time case) since the similarity transformation is symplectic.
One can then pose the problem of transforming Z by other
ities, say orthogonal,
symplectic similar_
so as to achieve various desirable numerical
properties or canonical forms.
presently being investigated.
This topic for further research is
-17-
4.3
Operation Counts, Timing, and Storage
th
We shall give approximate operation counts for the solution of n-order algebraic Riccati equations of the form (1) or (8).
Each operation
is assumed to be roughly equivalent to forming a + (b x c) where a,b,c
It is almost impossible to give an accurate
are floating point numbers.
operation count for the algorithm described above since so many factors
are variable such as the ordering of the RSF.
We shall indicate only a
ballpark 0(n3 ) figure.
Let us assume then that we already have at hand the 2n x 2n matrix
Z of the form (2) or (9).
Note, however, that unlike forming Z in
(2), Z in (9) requires approximately 4 n 3 additional operations to construct, given only F, G, and H.
This will turn out to be fairly negli-
gible compared to the counts for the overall process.
Furthermore, we
The
shall give only order of n3 counts for these rough estimates.
three main steps are:
Operations
(i)
(ii)
(iii)
reduction of Z to upper Hessenberg from
-(2n)
3
reduction of upper Hessenberg form to RSF
> 4k(2n)3
solution of XUll
=
4
3
n
U21
The number k represents the average number of QR steps required
per eigenvalue and is usually over-estimated by 1.5.
> 4k(2n)3
We write
since, in general, the reduction may need more operations
if ordering is required.
Using k = 1.5 we see that the total number
3
of operations required is at least 63 n . Should the ordering of the
RSF require, say, 25% more operations than the unordered RSF, we have
-18-
a ballpark estimate of about 75 n3 for the entire process.
Timing estimates for steps
(i) and (ii) may be obtained from [27]
for a variety of computing environments.
The additional time for
balancing and for step (iii) would then add no more than about 5% to
those times while the additional time for ordering the RSF is variable,
but typically adds no more than about 15%.
For example, adding 20% to
the published figures 127] for an IBM 370/165 (a typical medium speed
machine) under OS/360 at the University of Toronto using FORTRAN H
Extended with Opt. = 2 and double precision arithmetic, we can construct
the following table:
Riccati Equation
Order n =
10
20
30
40
CPU Time (Sec.)
0.2
1.3
4.0
9.0
In fact, these times are in fairly close agreement with actual observed
times for randomly chosen test examples of these orders.
Note the
approximately cubic behavior of time versus order.
Extrapolating these figures for a 64th order equation (see Example
5 in Section 6) one might expect a CPU-time in the neighborhood of
38 sec.
In fact, for that particular example the time was approximately
34 sec.
It must be re-emphasized here that timing estimates derived as
above are very approximate and depend on numerous factors in the actual
computing environment as well as the particular input data.
However,
such estimates can provide very useful and quite reliable information
if interpreted as providing essentially order of magnitude figures.
-19-
With respect to storage considerations the algorithm requires
8n
+ cn (c = a small constant) storage locations.
This fairly large
figure limits applicability of the algorithm to Riccati equations on
the order of about 100 or less in many common computing environments.
Of course, CPU time becomes a significant factor for n>100, also.
4.4
Stability and Conditioning
This section will be largely speculative in nature as very few
hard results are presently available.
A number of areas of continuing
research will be described.
With respect to stability, the implementation discussed in Section
4.1 consists of two effectively stable steps.
The crucial step is the
QR step and the present algorithm is probably essentially as stable as
QR.
The overall two step process is apparently quite stable numerically
but we have no proof of that statement.,
Concerning the conditioning of (1) (or (8)) almost no analytical
results are known.
The study of (1) is obviously more complex than
the study of even the Lyapunov equation
F X + XF + H = 0
where H = HT > 0.
(12).
And yet very little numerical analysis is known for
In case F is normal, a condition number with respect to in-
version of the Lyapunov operator FX = F X + XF is easily shown to
be given by
max
Xi(F) + Xi (F)|
i,j
min
Lj
(12)
i (F) + X (F)I
-20-
But in the general case, a condition number in terms of F rather than
F
0 I + I 0 FT ( 0
denotes Kronecker product) has not been determined.
Some empirical observations on the accuracy of solutions of certain
instances of (12) suggest that one factor influencing conditioning of
(12) is the proximity of the spectrum of F to the imaginary axis.
more specific, suppose F has an eigenvalue at a + jb withla
cally a < 0 is very small).
To be
>> 1 (typi-
t
l = 0 (10
) we lose approximately t digits
If
of accuracy and we might expect a condition number for the solution of
0 (1 0 t)
(12) to also be
in this situation.
There are some close connections between
(12) and (1) (and the
respective discrete-time versions) and we shall indicate some preliminary
observations here.
A perturbation analysis or the notion of a condition
number for (1) is intimately related to the condition of an associated
Lyapunov equation, namely one whose "F-matrix" approximates the closedloop matrix F-GX where X solves
where Y = Y
(1).
To illustrate, suppose X = Y + E
may be interpreted as an approximation of X.
0 = FT (Y+E) +
Then
(Y+E)F - (Y+E)G(Y+E) + H
T
T
(F-GY) E + E(F-GY) + (F Y + YF - YGY + H)
^AT
=F E + EF + H
where we have neglected the second-order term EGE.
Thus conditioning of
(1) should be closely related to nearness of the closed-loop spectrum
(c(F-GX)) to the imaginary axis.
made elsewhere;
Observations similar to these have been
see, for example, Bucy [31] where the problem is posed
as one of structural stability.
A condition number might, in some sense,
be thought of as a quantitative measure of the degree of structural
stability.
-21-
Another factor involved in the conditioning of
assumptions of stabilizability of
(F,B) and detectability of (C,F).
For example, near-unstabilizability of
or in a control energy sense
definitely causes
(1) relates to the
(i.e.,
(F,B) in either a parametric sense
near-singular controllability Gramian)
(1) to become badly
conditioned.
Our experience has
been that the ill-conditioning manifests itself in the algorithm by a
badly conditioned Ull.
Work related to the conditioning of
(1) and (8) is under continuing
investigation and will be the subject of another paper.
Such analysis
is, of course, independent of the particular algorithm used to solve
(1) or
(8), but is useful to understand how ill-conditioning can be
expected to manifest itself in a given algorithm.
-22-
5.
Advantages of the Schur Vector Approach and Further General Remarks
5.1
Advantages and Disadvantages of the Schur Vector Approach
The advantages of this algorithm over others using eigenvectors
as Potter's approach [10] and its extensions) are obvious.
(such
Firstly, the
reduction to RSF is an intermediate step in computing eigenvectors anyway (using the double Francis QR algorithm) so the Schur approach must,
by definition, be faster.
Secondly, and more importantly, this algorithm
will not suffer as severely from the numerical hazards inherent in computing
eigenvectors associated with multiple or near-multiple eigenvalues.
The
computation of eigenvectors is fraught with difficulties (see, e.g.
[21]
for
a cogent discussion) and the eigenvectors themselves are simply not needed.
All that is needed is a basis for the eigenspace spanned by the eigenvalues
of Z with negative real parts
time case).
ror)
(with an analogous statement for the discrete-
As good a basis as is possible
(in the presence of rounding er-
for this subspace can be found from the Schur vectors comprising the
matrixi 11 , independently of individual eigenvalue multiplicities.
reader is strongly urged to consult [32] and [21]
The
(especially pp. 609-610)
for further numerical details.
The fact that any basis for the stable eigenspace can be used to construct the Riccati equation solution has been noted by many people; see
or [3]
among others.
[12]
The main stumbling block with using the Schur vectors
was the ordering problem with the RSF but once that is handled satisfactorily
the algorithm is easy.
-23-
The Schur vector approach derives its desirable numerical properties
from the underlying QR-type process.
To summarize:
if you like the
eigenvector approach for solving the algebraic Riccati equation you must
like the Schur vector approach better.
Like the eigenvector approach, the Schur vector approach has the advantage of producing the closed-loop eigenvalues
(or whatever is appropriate
to the particular application from which the Riccati equation arises)
essentially for free.
And finally, an important advantage of the Schur
vector approach, in addition to its general reliability for engineering
applications, is its speed in comparison with other methods.
We have
already mentioned the advantage, by definition, over previous eigenvector
approaches but there is also generally an even more significant speed
advantage over iterative methods.
This advantage is particularly apparent
in poorly conditioned problems and in cases in which the iterative
method has a bad starting value.
Of course, it is impossible to make
the comparison between a direct versus iterative method any more precise
for general problems but we have found it not at all uncommon for an
iterative method, such as straightforward Newton [14], to take ten to
thirty times as long - if, indeed, there was convergence at all.
As mentioned above, a possible disadvantage of the method is the
storage requirement of at least two 2nx2n arrays.
Another disadvantage is
-1
the fact that for the computed X=U21Ull there is no guarantee of symmetry.
In practice, we have found the deviation from symmetry to be only slight for
most problems, becoming more pronounced only when the Riccati equation was
known to be "ill-conditioned."
This phenomenon might be used advantageously
if the deviation from perfect symmetry could be used to reliably monitor
conditioning.
-24-
5.2
Miscellaneous General Remarks
Remark 2:
An n
th-
order algebraic Riccati equation has a finite number
of solutions if the characteristic and minimal polynomials of F - GX are
equal where X is the unique nonnegative definite solution.
In that case there
are still as many as( nsolutions corresponding to as many as(
of the 2n eigenvalues of Z.
choices of n
Any of these solutions may also be generated by
the Schur approach, as for the eigenvector approach, by an appropriate
reordering of the RSF.
For most control and filtering applications we are
interested in the unique nonnegative definite solution and we have thus
concentrated the exposition on that particular case.
Remark 3: One of the most complete sources for an eigenvector-oriented
proof of Theorem 5 for the general case of multiple eigenvalues is
Martensson
[13].
But even a casual glance at that proof exposes the
awkwardness of fussing with eigenvectors and principal vectors.
The proof
using Schur vectors is extremely clean and easy by comparison and neatly
avoids any difficulties with multiple eigenvalues.
This
observation is but
one instance of the more general observation that Schur vectors can probably
always replace principal vectors
(or generalized eigenvectors) corresponding
to multiple eigenvalues throughout linear control/systems theory.
Principal
vectors are not generally reliably computable in the presence of roundoff
error anyway
(see [21]) and a basis for an eigenspace - but not the particular
one corresponding to the principal vectors - is all that is normally needed.
Use of Schur vectors will not only frequently provide cleaner proofs but is
also numerically much more attractive.
Remark 4:
The same Schur vector approach employed in this paper can also
be used instead of the eigenvector approach for the nonsymmetric matrix quadratic
-25-
equation
(13)
XEX + FX + XG + H = 0
where E
I
mxn
, F £ IR
nxn
,GE R
mxm
,H C I
nxm
, andXE R
nxm
.
In this case,
we work with the (m+n) x (m+n) matrix
Z
-GEF
and various solutions of (13) are determined by generating appropriate
combinations of m eigenvalues of Z along the diagonal of the RSF of Z.
corresponding m Schur vectors give the solution X = U
Ul
IR
~mxm
, U21 C IR
nxm
.
[33],
-1
U1 as before where
21 11
The analogous remarks apply for the corresponding
nonsymmetric "discrete-time equation."
both cases.
The
Proofs are essentially the same in
Further details on the eigenvector approach can be found in
[34].
Remark 5:
Special cases of the matrix quadratic equations such as (1),
(8), or (13) include the Lyapunov equation
part FTXF - X + H = O)
(12)
(or its discrete-time counter-
and the Sylvester equation
FX + XG + H = 0
(14)
(or its discrete-time counterpart FXG - X + H = 0).
Thus setting an appropriate block of the Z matrix equal to 0 provides a
method of solving such "linear equations" and, in fact, this method has even
been proposed in the literature
[35].
However, the approach probably has
little to recommend it from a numerical point of view as compared to applying
the Bartels-Stewart algorithm [39]
and we mention it only in passing.
-26-
6.
Examples
In this section we give a few examples both to illustrate various
points discussed previously and to provide some numerical results for
comparison with other approaches.
All computations were done at
M.I.T. on an IBM 370/168 using FORTRAN H Extended
(Opt. = 2) and
double precision arithmetic.
Example 1:
The Schur vector approach is obviously not well-suited to
hand computation - which partly explains its desirable numerical
properties.
However, to pacify a certain segment of the population
a "hand example" is provided in complete detail.
T
Consider the equation
-1T
A X + XA - XBR
B X + Q = 0
(15)
which arises in a linear-quadratic optimal control context with
1,
A=(
B=
A,
Then
0
1
0
0
0
0
0
-1
0
0
0
-2
-1
0
-1
0
and the matrix
1
_
_V5
2
10
1
__5
2
10
21
305
2
10
_1
2
-
_3V/
10
3/5
1
10
2
3_5
1
10
2
1
10
2
10
1
2
R= 1,
Q=
-27-
is an orthogonal matrix which reduces Z to RSF
-1
0
1
0
-1
-1
1
O
0
1
0
0
0
0
1
S = U ZU =
Then the unique positive definite solution of (15) is given by the
solution of the linear matrix equation
XUll = U21
or
11
2
2
1lO
0
10
10
Thus X
=
( )
2
10
and it can quickly be checked that the spectrum of
the "closed-loop matrix"
is {-1, -1} as was
(A- BR-1BTX) =
\-1
-2
evident from Sll.
Example 2:
For checking-purposes consider the solution of (15) with the
following uncontrollable but stabilizable, and unobservable but detectable data:
A= 6
3)
B(
)
R=l,
(
)
-28-
/9c
The solution of (15) is X =
-
loop spectrum is
}.
6c
where c = 1 + W and the closed4c
6c
These values were all obtained correctly
to at least 14 significant figures as were the values for the corresponding discrete-time problem
T
T
T
-lT
A XA - X - ATXB(R+ B XB)
B XA + Q = 0
(16)
the solution of which is
9d
6d\
X=
\6d
4d
1+
where d -
2
Example 3:
1
and the closed-loop spectrum is {-
2' 2
/0~.9512
0~
,
4.877
4.877
1.1895
3.569/
I
B=
0
0.9048
3--
.005
,
R=
The solution of
0
Q
3
X
}
For further comparison purposes consider the discrete-time
system of Example 6.15 in [36] where
A
3 -
-2'
0
0.02)
(16) is given by
0.010459082320970
0.003224644477419
0.003224644477419
0.050397741135643/
=
and the feedback gain F =
0.071251660724426
F3569839235296
.013569839235296
(R+ B XB)
BTXA is given by
-0.070287376494153
0.045479287667006/
0.045479287667006/
-29-
Note the typographical error in the
(1,2)-element of F in [36].
The
closed-loop eigenvalues are given by
0.508333461684191 and 0.688069670988913
These are definitely different from [36]
but have the same sum.
Our
numbers do appear to be the correct ones.
Example 4:
We now consider somewhat higher order Riccati equations
arising from position and velocity control for a string of high-speed
vehicles.
The matrices are taken from a paper by Athans, Levine, and
Levis [37].
For a string of N vehicles it is necessary to solve the
Riccati equation
ATX + XNA N - X NBNR1B+
R
N Q
NN
NNN
NNN
NNN
0
where all matrices are of order n = 2N-1 and are given by
A11
A12
A22
A23
N
N-2,N-2
A
N-2,N-l
,N-1
0
where A,k
Akk
'
Akkl=
=
+
0
--1
-1
-30-
and B
1
=
NBN
N
BT
diag{l,0,,0,1
N,=
QN
..
0,.,1}
diag{0,10,0,10,...,10,0}
.
For the case of 5 vehicles we repeated the calculations presented
in [373.
The correct values for X rounded to six significant figures
are:
1.36302 2.61722 -0.705427 0.936860 -0.293666 0.477354 -0.197375 0.211212 -0.166552
7.59255 -1.68036
1.47522
-0.459506 0.665147 -0.266142 0.280654 -0.211212
1.77478
2.15771
-0.609136 0.670717 -0.262843 0.266142 -0.197375
8.25770
-1.94650
1.75587
-0.670717 0.665147 -0.477354
1.80560
1.94650
-0.609136 0.459506 -0.293666
8.25770
-2.15771
1.47522
-0.936860
1.77478
1.68036
-0.705427
7.59255
-2.61722
[SYMMETRIC]
\
1.36302
While 4 or 5 decimal places are published in [37],
it can be seen that,
surprisingly, only the first and sometimes the second were correct.
Sub-
stitution of our full 16 decimal place solution into the Riccati equation
-14
gives a residual of norm on the order of 10
(-consistent with a condition estimate of U11 of 26.3) while the residual for the solution in
[37] has a large norm on the
order of
10
.
The
closed-loop
eigenvalues for the above problem (again rounded to six significant
figures) are:
-1.00000
-1.10779
-1.45215
-1.67581
-1.80486
+
+
+
+
0.852759
1.26836
1.51932
1.66057
j
j
j
j
We also computed the Riccati solution and closed-loop eigenvalues
for the cases of 10 and 20 vehicles.
This involved the solutions of
19th and 39th order Riccati equations, respectively, and rather than
-31-
reproduce all the numbers here we give only the first five and last
five elements of the first row (or column) of X and the fastest and
slowest closed-loop modes.
Again all values are rounded to just six
significant figures; the complete numerical solutions are available
from the author.
First row (column) of Riccati
Solution
N=10
N=20
n=19
n=39
1.40826
1.42021
2.66762
2.68008
-0.658219
-0.646127
1.04031
1.06539
-0.242133
-0.229761
-0.0515334
-0.0123718
0.103453
0.0250824
-0.0472086
-0.0120915
0.0504036
0.0124632
-0.0452352
-0.0119545
Fastest and Slowest Closed-Loop Modes
N=10
n=19
N=20
n=39
-1.83667
-1.84459
+ 1.69509 j
-0.862954
+ 1.70368
j
-0.662288
+ 0.494661 j
The closed-loop eigenvalues for the case of, say, 10 vehicles interlace
andinclude, as a subset, those of 5 vehicles.
Similarly, those for 20
vehicles interlace and include, as a subset, those of 10 (and hence 5)
vehicles.
It appears evident that both the elements of the Riccati
solution and the closed-loop eigenvalues are converging to values in
some finite region.
Example 5:
This example involves circulant matrices.
solve (15) with
We wish to
-32-
-2
1
and BR
B
1
0.
1
0 . . .1
I, Q
I
*2
The matrices A, BR
the Riccati solution X e JR
X0
Xn-l
Xn
X1
X
Xn-l
X1
X0
X= X 2
2
is
B,
Q are all circulant so
known to be circulant of the form
. . . x
X2
In fact, there is a simple transformation which "diagonalizes" the
Riccati equation and allows the solution of (15) to be recovered via
the solution of n scalar quadratic equations and an inverse discrete
Fourier transform.
The details of this procedure and related analysis
of circulant systems can be found in the work of Wall
r[38].
For this
example, we have n = 64 and the xe are given by
k=OX 4 2+2cos
5-4cos
where w64 is a 64-th root of unity.
0
+ 4
26k
ik
The solution was computed by the
Schur vector approach and checked by means of the circulant analysisf
-33-
of Wall. Our
figures.
computed Riccati solution had at least 13 significant
For reference purposes we list
Xll = 0.37884325313566
x12 = 0,18581947375535
X44 = 0.37884325313567
x
= 0.18581947375536
The closed-loop eigenvalues are all real and are arranged as follows:
-4.1231056256177
-4.1137632861146
-4.1137632861146
31 eigenvalues of multiplicity 2
-0.99999999999991
th
This 64- order example required approximately 50 sec. of CPU time on
the 370/168 at M.I.T. and approximately 34 sec. on the 370/165 at the
University of Toronto - both using FORTRAN H Extended (Opt. =2),
double
precision.
Example 6:
This example is one which would be expected to cause problems
on physical grounds and which appears to give rise to an "ill-conditioned
Riccati equation".
Consider the solution of (15) with
-34-
0
1
o
Oi
0
0
A=01
,
B
O/
O0
Q = diag{q,0,...,0},
R=r
Here we have a system of n integrators connected in series.
It is desired'
th
to apply a feedback controller to the n--system (which is to be integrated
n times) so as to achieve overall asymptotic stability.
x1
th
Only deviations of
integral of the constant system) from 0 are penalized.
(the n-
The
reachability Gramian
t
Wt =
f
;A
e
T sAT
BB e
ds
while positive definite for all t>0, becomes more nearly singular as n
increases.
The system ismhard to control" in the sense of requiring a
large amount of control energy
(as measured by
I Wt lI).
The closed-loop eigenvalues are easily seen to be the roots of
x2n +
(_l)n q = 0
r
with negative real parts.
pattern.
In
These eigenvalues lie in a classic Butterworth
It can also be easily verified that
r
= product of the closed-loop eigenvalues
-35-
We attempted the solution of (15) with the above matrices and q= r= 1.
While the closed-loop eigenvalues were determined quite accurately as expected
(approximately 14 decimal places using IBM double precision), the
Riccati solution was increasingly less accurate as n increased due to the
increasingly ill-conditioned nature of Ull.
For example, for n = 21 there
was already a loss of 10 digits of accuracy (consistent with a condition
estimate of 0(10
10)
for Ull) in xl
(=1).
Other computed elements of X were
as large as 0(109) in magnitude.
Repeating the calculations with q = 10 , r = 1 there was a loss of
approximately 12 digits of accuracy in xln (=100) for n= 21.
other elements of X were as large as 0(10 11) in magnitude.
In this case
Again, the closed-
loop eigenvalues were determined very accurately.
Our attempts to get Newton's method to converge on the above problem
were unsuccessful.
Obviously, there is more that can be said analytically about this
problem.
Our interest here has been only to highlight some of the numerical
difficulties.
-36-
7.
Concluding Remarks
We have discussed in considerable detail a new algorithm for solving
algebraic Riccati equations.
A number of numerical issues have been
addressed and various examples given.
The method is apparently quite
numerically stable and performs reliably on systems with dense matrices
of up to order 100 or so, storage being the main limiting factor.
For some reason, numerical analysts have never really studied algebraic Riccati equations.
The algorithm presented here can undoubtedly
be refined considerably from a numerical point of view but it nonetheless
represents an immense improvement over algorithms heretofore proposed.
Some topics of continuing research in this area will include:
(i)
conditioning of Riccati equations,
(ii)
use of software to sort blocks of the RSF diagonal into
just the two appropriate groups rather than within the
two groups as well,
(iii) making numerically viable the use of symplectic transformations such as in 1171 to reduce the Hamiltonian
or
symplectic matrix Z to a convenient canonical form.
Each of these topics is of research interest in its own right in addition
to the application to Riccati equations.
-37-
8.
References
[1]
Wonhamn,, W.M., On a Matrix Riccati Equation of Stochastic Control,
SIAM J. Contr., 6(1968), 681-697.
[2]
Reid, W.T., Riccati Differential Equations, Academic Press, New
York, 1972.
[31 Willems, J.C., Least Squares Stationary Optimal Control and the
Algebraic Riccati Equation,
IEEE Trans. Aut. Contr., AC-16(1971),
621-634.
[4]
Silverman, L.M., Discrete Riccati Equations: Alternative Algorithms,
Asymptotic Properties, and System Theory Interpretations, in
Advances in Control Systems, Vol. 12, (Leondes, Ed.), Academic Press,
New York, 1976, pp. 313-386.
[51 Lainiotis, D.G., Partitioned Riccati Solutions and Integration-Free
Doubling Algorithms, IEEE Trans. Auto. Contr., AC-21 (1976),
677-689.
[61 Rodriguez-Canabal, J., The Geometry of the Riccati Equation,
Stochastics, 1(1973), 129-149.
7]1 Pachter, M., and T.E. Bullock, Ordering and Stability Properties
of the Riccati Equation, Na. Res. Inst. for Math. Sci. Report,
WISK 264, Pretoria, June 1977.
181 Von Escherich, G., Die Zweite Variation der Einfachen Integrale,
Wiener Sitzungsberichte, 8(1898), 1191-1250.
[91 MacFarlane, A.G.J., An Eigenvector Solution of the Optimal Linear
Regulator Problem, J. Electron. Contr., 14(1963), 643-654.
[101 Potter, J.E., Matrix Quadratic Solutions, SIAM J. Appl. Math., 14(1966),
496-501.
1111 Vaughn, D.R., A Nonrecursive Algebraic Solution for the Discrete
Riccati Equation, IEEE Trans. Auto. Contr., AC-15(1970), 597-599.
112] Fath, A.F., Computational Aspects of the Linear Optimal Regulator
Problem, IEEE Trans. Auto. Contr., AC-14(1969), 547-550.
[131 MArtensson, K., New Approaches to the Numerical Solution of Gptimal
Control Problems, Lund Institute of Technol., Divis. of Auto. Contr.,
Report No. 7206, Lund, Sweden, Mar. 1972.
1141 Kleinman, D.L., On An Iterative Technique for Riccati Equation Computations, IEEE Trans. Auto. Contr., AC-13(1968), 114-115.
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[15]
Farrar, F.A., and R.C. DiPietro, Comparative Evaluation of Numerical Methods for Solving the Algebraic Matrix Riccati Equation,
United Technologies Research Center Report No. R76-140268-1, East
Hartford, CT, Dec. 1976.
[16] Hewer, G.A., and G. Nazaroff, A Survey of Numerical Methods for
the Solution of Algebraic Riccati Equations, Naval Weapons
Center Report, China Lake, CA.
[17]
Laub, A.J., and K.R. Meyer, Canonical Forms for Hamiltonian and
Symplectic Matrices, Celestial Mechanics, 9(1974), 213-238.
[18]
Laub, A.J., Canonical Forms for a-Symplectic Matrices, M.S. Thesis,
School of Mathematics, Univ. of Minnesota, Dec. 1972.
[19]
Stewart, G.W., Introduction to Matrix Computations, Academic Press,
New York, 1973.
[20] Murnaghan, F.D., and A. Wintner, A Canonical Form for Real Matrices
Under Orthogonal Transformations, Proc. Na. Adad. Sci., 17(1931),
417-420.
[21]
Golub, G.H., and J.H. Wilkinson, Ill-Conditioned Eigensystems and
the Computation of the Jordan Canonical Form, SIAM Rev., 18(1976)
578-619.
[22]
Laub, A.J., A Schur Method for Solving Algebraic Riccati Equations,
LIDS Report LIDS-R-859, Laboratory for Information and Decision
Systems, Massachusetts Institute of Technology, Cambridge, MA 02139.
[23]
Dorato, P., and A. Levis, Optimal Linear Regulators:
The DiscreteTime Case, IEEE Trans. Auto. Contr., AC-16(1971), 613-620.
[24] Pappas, T., A.J. Laub, and N.R. Sandell, On the Numerical Solution of
the Discrete-Time Algebraic Riccati Equation, LIDS Report, LIDS-P-908,
Laboratory for Information and Decision Systems, Massachusetts
Institute of Technology, Cambridge, MA 02139.
[25] Anderson, B.D.O., and J.B. Moore, Optimal Filtering, Prentice-Hall,
Englewood Cliffs, NJ, 1979.
[26]
Fortran Subroutines for Calculating
Stewart, G.W., HQR3 and EXCHNG:
and Ordering the Eigenvalues of a Real Upper Hessenberg Matrix, ACM
Trans. Math. Software, 2(1976), 275-280.
[27] Smith, B.T., et. al., Matrix Eigensystems Routines -- EISPACK Guide,
Second Edition, Lect. Notes in Comp. Sci., Vol. 6, Springer-Verlag,
New York, 1976.
[28] Forsythe, G.E., M.A. Malcolm, and C.B. Moler, Computer Methods
for Mathematical Computations, Prentice-Hall, Englewood Cliffs,
NJ, 1977.
-39-
[29] Dongarra, J.J., J.R. Bunch, C.B. Moler, and G.W. Stewart, LINPACK
User's Guide, SIAM, Philadelphia, 1979.
[30] Parlett, B.N., and C. Reinsch, Balancing a Matrix for Calculation
of Eigenvalues and Eigenvectors, Numer. Math., 13(1969), 296-304.
[31] Bucy, R.S., Structural Stability for the Riccati Equation, SIAM
J. Contr., 13(1975), 749-753.
[321 Wilkinson, J.H., The Algebraic Eigenvalue Problem, Oxford University
Press, London, 1965.
[33] Coppel, W.A., Matrix Quadratic Equations, Bull. Austral. Math. Soc.,
10(1974), 377-401.
[34] Meyer, H.-B., The Matrix Equation AZ+B-ZCZ-ZD=0, SIAM J. Appl.
Math, 30(1976), 136-142.
[35] Bar-Ness, Y., and G. Langholz, The Solution of the Matrix Equation
XC-BX=D as an Eigenvalue Problem, Int. J. Sys. Sci., 8(1977), 385392.
[36] Kwakernaak, H., and R. Sivan, Linear Optimal Control Systems,
Wiley, New York, 1972.
[37] Athans, M., W.S. Levine, and A. Levis, A System for the Optimal and
Suboptimal Position and Velocity Control for a String of High-Speed
Vehicles, Proc. 5th International Analogue Computation Meetings,
Lausanne, Switzerland, Sept. 1967.
[38] Wall, J.E., Control and Estimation for Large-Scale Systems Having
Spatial Symmetry, Ph.D. Thesis, M.I.T., Aug. 1978; Electronic
Systems Lab. Rept. ESL-TH-842.
[39] Bartels, R.H., and G.W. Stewart, Solution of the Matrix Equation
AX + XB = C, Comm. ACM, 15(1972), 820-826.
-47-
APPENDIX 1
We outline here how to set up the "symplectic approach" when the
matrix F in
FTXF - X - F XG
I
is singular.
(G 2 +Gl XG) -
G1 X F + H = O
All other assumptions and notation of Section 3 will
be the same.
Letting xk denote the state at time tk and Xk the corresponding
adjoint vector, recall the Hamiltonian difference equations arising
from the discrete maximum principle:
(I
G
(xF
0
(
Xk+l
Note that if F were invertible we could work with the symplectic
matrix
F
H
which is just (9).
I
-F -TH
Here, instead, we shall be concerned with a "symplectic
generalized eigenvalue problem"
Lz = XMz
with
F
L = -H
F-
0
I
G)
I)
O
F
-48-
and symplectic in the sense that if X 4 0 is a generalized eigenvalue
then
X
is a generalized eigenvalue.
In fact, L and M are characterized
by the property that
LJLT = MJMT
where
J =(C
In our specific situation LJL
)
= MJM
=
FT
0
There is even more "reciprocal symmetry" in the problem.
With F
singular there must be least one generalized eigenvalue at 0 and to each
such generalized eigenvalue there corresponds its reciprocal at a.
The
generalized eigenvalues can then be arranged in two groups of n as before:
1
n
with O<Ixil<l.
1
- __
We then find a basis for the generalized eigenspace cor-
responding to 0,...,O,X ,...Xk and proceed essentially as before.
details are omitted here as they are the subject of a forthcoming
paper with T. Pappas.
The
-49APPENDIX 2
In this appendix we provide FORTRAN source listings for one possible
implementation of the Schur vector approach described in the paper.
Sub-
routines for solving both the continuous-time algebraic Riccati equation
(1) tRICCND] and the discrete-time algebraic Riccati equation (8) [RICDSDJ
are given.
The subroutine names are derived from the following nomenclature
convention for a family of subroutines to solve Riccati and various other
matrix equations:
subroutine name:
where
XXXYYZ
(RIC
XXX =
Riccati equation
LYP
Lyapunov equation
SYL
Sylvester equation
ICN
continuous-time version
YY =
DS
discrete-time version
single (short) precision version
Z=
D
double (long) precision version
Subroutine RICCND calls or further requires the following additional
subroutines:
BALANC, BALBAK, DDCOMP, DSOLVE, EXCHNG, HQR3, MLINEQ, ORTHES,
ORTRAN, QRSTEP, SPLIT
Subroutine RICDSD requires each of the 11 subroutines above as well as
the two additional subroutines MULWOA, MULWOB.
-50-
All the additional subroutines required have also been listed here
with the exception of BALANC, BALBAK, ORTHES, and ORTRAN which are
available in EISPACK [27].
These subroutines are being used in the environment described in
Section 6 as part of a package called LQGPACK.
This package is a
preliminary version of a set of subroutines being developed at M.I.T.'s
Laboratory for Information and Decision Systems to solve linear-quadraticGaussian control and estimation problems.
The package has also been run
in a single precision version on a CDC 6600.
However, at this time we
make no claims of portability of the code to other machines.
The
code listed here is solely for illustrative purposes.
Finally, we add two additional technical notes:
NOTE 1:
A fairly reliable estimate of the condition number of U11 with
respect to inversion is returned by RICCND or RICDSD in WORK (1).
NOTE 2:
The subroutine HQR3 contains a small error which can occasionally
cause RICCND or RICDSD to give erroneous or misleading information.
The
trouble arises when ORTHES produces an upper Hessenberg form with a zero
on the first subdiagonal.
HQR3 then correctly orders the resulting RSF
both above and below that zero element but not necessarily globally.
In practice this almost never happens and it has only ever been observed
for certain low-order examples with all coefficient matrices diagonal.
This error in HQR3 can and will be corrected.
In the interim,
error can either be ignored (a safe strategy for virtually all "real
problems") or temporarily patched by the following scheme.
the
-51Let ai+l
i
be a zero element of the upper Hessenberg matrix A
(the output of ORTHES).
replaced by e
Then before HQR3 is called, ai+l
i
should be
(-Iai,if
+ jai+l,i+ll)where e is the machine precision
(EPS) defined by
e = min {6: f(l+I6 I) f
1}
(f (-) denotes floating point operation).
The source listings now follow.
RICCND 1
-52-
SUBROUTINE RICCND
+
C
C
*****A RAE TER S:
INTEGER NZ, NF,NG,NH,N,NN,ITYPE(NN),IPVL (NN) ,IPVS (N)
DOUBLE PRECISION Z(NZ,NN),W(NZ,NN),F(NF,N),G(NG,N),H(NH,N),
EF (NN),EI(NN),WCRK(N),SCALE(NN)
+
C
C
j*****.':LOCAL VARIABLES:
INTEGER I,J,LOW,IGH,NLOW,NUP
DOUBLE PRECISION EPS, EPSP1,ZNOR.i,,T,ALPHA,COND
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
c
C
C
C
C
C
C
C
c
C
C
C
C
C
(NZ,NF, NG, NH, N,NN,Z,W,F,G,H,ER,EI,ORK,
SCALE,ITYPE,IPVL,IPVS)
*****FUNCTICNS:
DOUBLE PRECISION DABS,DSQRT
*****SUBROUTINES CALLED:
BALA.NC,BALBAK,HQR3, MLINEQ,ORTHES,ORTRAN
--
::::::::::..:R1C009:0:::
'
::::
*****PURPOSE:
THIS SUBROUTINE SOLVES THE CONTINUOUS-TIME
ALGEBRAIC MATRIX RICCATI EQUATION
T
F *X
+ X*F -
X*G*X +
H =
0
RIC00010
RIC00020
RICO0030
RIC00040
RICO0050
RICOOO60
RIC00070
RICOOO380
RIC00090
RICOOo00
RICOOl110
RIC00120
RIC00130
RIC00140
RICO 1 5O
RIC00160
RICOO1 70
RIC0so80
0:
:
RIC00200
RIC00210
RIC0o0220
RIC00230
RI CO 2 40
RIC00250
RIC00260
RIC00270
APPROACH.
HAMILTONIAN-EIGENVECTOR
THE
BY LAUB'S VARIANT OF
RIC00280
R C00290
*****PAR$AMETER DESCRIPTION:
RIC00 300
ON INPUT:
RIC00310
RIC00320
ROW DIMENSIONS OF THE ARRAYS CONTAINING
NZ,NF,NG,NH
RIC00330
AS
RESPECTIVELY,
H,
AND
Z (AND W),F,G,
RIC00340
DECLARED IN THE CALLING PROGRAM DIMENSION
RIC00350
STATEMENT;
RIC00360
RIC00370
F,G,H;
MATRICES
OF
THE
ORDER
N
RIC00380
RIC00390
= 2*N = ORDER OF THE INTERNALLY GENERATED
NN
RIC00400
W;
Z
AND
MATRICES
RIC00410
RIC00420
MATRIX;
(REAL)
X
N
AN
N
F
RIC00430
RIC00440
DEFINITE
NONNEGATIVE
SYMMETRIC,
X
N
N
G,H
RIC00450
(REAL) MATRICES.
RIC00460
R-IC00470
ON OUTPUT:
RIC00480
AN N X N ARRAY CONTAINING THE UNIQUE POSITIVE :RIC00490
H
RICOO500
(OR NONNEGATIVE) DEFINITE SOLUTION OF THE
RIC00510
RICCATI EQUATION;
RIC00520
REAL SCRATCH VECTORS OF LENGTH 2*N; ON OUTPUT RIC00530
ER,EI
RIC00540
(ER(I),EI(I)), I=1,N CONTAIN THE REAL AND
RIC00550
IMAGINARY PARTS, RESPECTIVELY, OF THE N
-53-
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
CLOSED LOOP EIGENVALUES
SPECTRUM OF F - G*X);
5
C
C
C
10
20
RICCND 2
(I.E.,
THE
RIC00560
RIC00570
RIC00580
Z'i
2*N X 2*N REAL SCRATCH ARRAYS USED FOR
RIC00590
COMPUTATIONS INVOLVING THE HAMILTONIAN
RICOO600
MATRIX ASSOCIATED WITH THE RICCATI EQUATION;
RIC00610
RIC00620
WORK,SCALE
REAL SCRATCH VECTORS OF LENGTHS N, 2*N,
RIC00630
RESPECTIVELY;
ON OUTPUT, WORK(1) CONTAINS A RIC00640
CONDITION NUMBER ESTIMATE FOR THE FINAL NTH
RIC00650
ORDER LINEAR MATRIX EQUATION SOLVED;
RIC00660
RIC00670
ITY PE,IPVL, IP VS
INTEGER SCRATCH VECTORS OF LENGTHS 2*N, 2*N,
RIC00680
N, RESPECTIVELY.
RIC00690
RIC00700
***NOTE:
ALL SCRATCH ARRAYS MUST BE DECLARED AND INCLUDED
RIC00710
IN THE CALL.***
RIC00720
RIC007 30
*****ALGORITHM NOTES:
RIC00740
IT IS ASSUMED THAT G AND H ARE NONNEGATIVE DEFINITE AND THAT (F,B) RIC00750
T
RIC00760
RIC00770
=G
IS STABILIZABLE AND (C,F) IS DETECTABLE WHERE B*B
T
RIC00780
RIC00790
AND C*C = H (C OF FULL
(B OF FULL RANK = RANK (G))
RANK = RANK (H))
IN WHICH CASE THE SOLUTION (RETURNED IN THE
RIC00800
RIC00810
ARRAY H) IS UNIQUE AND NONNEGATIVE DEFINITE.
RIC00820
*****HISTORY:
RIC00830
WRITTEN BY ALAN J. LAUB (ELEC. SYS. LAB.., M.I.T., RM. 35-331,
RIC00840
CAMBi:IDGE, MA 02139, PH.: (617) - 253-2125), SEPTEMBER 1977.
RIC00850
MOST RIlCENT VERSION:
SEP. 15, 1978.
RIC00860
RI C00870
.:::
: 1::::::880::::
RIC00890
EPS IS AN INTERNALLY GENERATA.D MACHINE DEPENDENT PARAMETER
RIC00900
SPECIFYING THE RELATIVE PRECISION OF FLOATING POINT ARITHMETIC.
RIC00910
FOR EXAMPLE, EPS = 16.ODO**(-13) FOR DOUBLE PRECISION ARITHMETIC
RIC00920
ON IBM S360/5370.
RIC00930
RIC00940
EPS= .OD0
RIC00950
EPS=0.5D0*EPS
RIC00960
EPSP =EPS+1.O D
RIC00970
IF (EPSPl.GT.1.0DO) GO TO 5
RIC00980
EPS=2.ODO*EPS
RIC00990
I1C01000
SET UP HAMILTONIAN MATRIX
RIC01010
RIC01 020
RIC01030
DO 20 J=1,N
DO 10 I=1,N
RIC01040
RIC0 1050
Z (I ,J) =F (I,J)
Z (N+I,J)=-d (I,J)
Rt C01 060
RICO 1 70
Z (I ,N+J) -G (I,J)
RIC01080
Z (N+I,N+J)=-F(J,I)
CONTINUE
RIC01090
CONTINUE
RIC01100
-54-
C
C
C
BALANCE
Z
CALL BALANC
C
C
C
30
40
C
C
C
C
C
C
C
50
60
C
C
C
C
COAPUTE
(NZ,NN,Z,LOW,IGH,SCALE)
1-NORM OF Z
ZNORM=. ODO
DO 40 J=1,NN
T=0.ODO
DO 30 I= 1,NN
T-=T+DABS (Z (I ,J) )
CONTINUE
IF (T.GT.ZNGRM) ZNORM=T
CONTINUE
ALPHA=DSQRT (ZNORM) + 1. 0DO
-1
COMPUTE W; = (ALPHA*I + Z)
* (ALPHA*I - Z), AN ANALYTIC FUNCTION
OF Z MAPPING THE LEFT HALF PLANE TO THE EXTERIOR OF THE UNIT
DISK. THIS PERMITS DIRECT APPLICATION OF HQR3.
THIS STEP MAY
BE REMOVED IF HQR3 IS MODIFIED APPROPRIATELY.
DO 60 J=1,NN
DO 50 T=1,NN
W(I, J) =-Z (I, J)
CONTINUE
W (3,3) =ALPHA+W (J,J)
Z (J,J)=ALPHA+Z (J,J)
CONTINUE
CALL MLINEQ (NZ, NZ,NN,NN,Z,W,COND,IPVL, ER)
REDUCE W TO REAL SCHUR FORM WITH EIGENVALUES OUTSIDE THE UNIT
DISK IN THE UPPER LEFT N X N UPPER QUASI-TRIANGULAR BLOCK
N LO W_ 1
NUP=NN
CALL ORTHES (NZ, NN,NLOW,NUP,W,ER)
CALL ORTRAN (NZ,NN,NLOW, NUP,W,ER,Z)
DO 15 I=2,NN
IF(W(,I-1).EQ. 0.0D ) W (I,I-1)= .OD-14
15 CONTINUE
CALL HQE3 (W,Z,NN,NLOW,NUP,EPS,ER,EI,ITYPE, NZ,NZ)
C
C
C
C
70
80
RICCND 3
COMPUTE SOLUTION OF THE RICCATI EQUATION FROM THE ORTHOGONAL
MATRIX NOW IN THE ARHAY Z.
STORE THE RESULT IN THE ARRAY H.
CALL BALBAK (NZ,NN,LOW,IGH, SCALE, NN,Z)
DO 80 J=1,N
DO 70 I1=,N
F (I J) =Z (J,I)
H (I ,J) =Z (N+J,I)
CONTINUE
CONTINUE
CALL MLINEQ (NF,NH,N,N,F,H,COND,IPVS,WORK)
RIc01110
RIC 1 1 20
RIC01 30
RIC0 1 140
RICO 150
RIC01160
RIC01170
RIC01180
RICO 1190
RIC01200
RIC01210
RIC01220
RIC01230
RIC01240
RIC01250
RIC01260
RIC01270
RIC01280
RIC01290
RIC01300
RIC01310
RIC 01320
RIC01330
RIC01340
RIC01350
RICO 360
RIC01370
RIC01 380
RIC0 1390
RIC01400
RIC01 410
R IC01 42 O
RIC03 1430
RIC01440
RIC01450
RI CO 14 60
RIC01 470
RIC01480
RIC01490
RIC01500
C01 51 0
RIC01520
RIC01530
RIC01540
RICO1550
RIC01560
RIC01570
RIC01580
RIC01590
RIC01600
RIC01610
RIC01620
RIC01630
RIC0 1640
RIC01650
-55-
RICCND 4
WORK (1) =COND
C
C
44400
TTRANSFORM
BACK TO GET THE CLOSED LOOP SPECTRUM
C
DO 110 I=1,N
IF (ITYPE(I).GE.0) GO TO 90
WRITE (6,t44400) I
FORMAT (lX, I4, X,41HHTH EIGENVALUE NOT SUCCESSFULLY CALCULATED)
RETUJRN
90
100
110
RIC0174O
IF (ITYPE (T).GT.0) GO TO 100
ER (I) =ALPHA* ( . OD-ER (I ))/(1. ODO+ER (I))
EI (I) =0.OD0
GO TO 110
IF
(ITYPE(I).EQ.2) GO TO 110
T=ALPHA/ ((1. ODO+ER (I)) **2+El (I) **2)
ER (I)= (1. ODO-ER (I) **2-EI (i) **2) *T
EI (I) =-2. ODO*EI (I) *T
ER (I+1) =ER(I)
El (I + 1)=-EI (I)
CONTINU E
RETURN
RIC 1750
RIC01760
RIC01770
RIC01780
RIC01790
RIC01800
RIC018
s
RIC01820
RIC01830
RICO 1840
RIC01 850
RICO 1860
LAST LINE OF RICCND
RIC01880
RIC01890
RIC01900
g
C
C
RIC0 1660
RIC01 670
RIC01680
RIC01690
RICO1700
RIC01710
RIC01720
RIC01730
RIC0t870
END
RICDSD 1
-56-
SU;i3ROUTINE RICDSD
+
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
c
C
C
C
C
C
C
C
C
C
C
C
C
(N ,NFNGNH,N, NN,Z,,F,G,H,EElEI,WORK,
SCALE, ITYPE, IPVT)
*****P ARAETERS:
INTEGER NZ,NF,NG,NH N,NNITYPE (NN),IPVT(N)
DOUBLE PRECISION Z(.NZ,NN),W(NZ,NN),F(NF, N),G(NG,N),H(NH,N) ,
ER (NN) ,EI(NN) ,WORK (N) ,SCALE (NN)
+
VARIABLES:
*****LOCAL
INTEGER I,J,K,LOW,IGH,NLOW,NUP
DOUBLE PRECISION EPS,EPSP1,CCNDCON
NDP1
**$***SUBlROUTINES CALLED:
BALANC, BALBAK,DDCOMP,DSOLVE, IQR3,MLINEQ,MULWOA,,MULWOB,
ORTHES,ORTRAN
:::::-::::
..... :
::::':::::.
.::::
RICOOO1(
RIC0002(
RIC0003(
RIC 0004(
RIC0005(
RIC0006(
RIC0007(
RICOO08(
RICO09(
RICOa O(
RIC0011(
RIC0012(
RICO 13(
RICOO 14(
R IC00 15(
RIC00!6(
:':
::::'::''::* RICO017(
RICo018(
RIC00 19(
*****PURPOSE:
RICOO20(
THIS DOUBLE PRECISION SUBROUTINE SOLVES THE DISCRETE-TIME
RIC002V1
ALGEBRAIC MATRIX RICCATI EQUATION
RIC0022(
RIC0023(
T
-1
T
T
T
RIC0024(
)*G1 *X*F + H
X = F *X*F - F *X*GI*((G2 + G1 *X*G1)
RIC0025(
RIC0026(
BY LAUB S VARIANT OF THE HAMILTONIAN-EIGENVECTOR APPROACH.
Gl
AND
RIC0027i
MATRICES
AND
THE
THE MATRIX F IS ASSUMIED TO BE NONSINGULAR
G2 ARE ASSUMED TO BE COMBINED INTO THE SQUARE ARRAY G AS FOLLOWS: RIC0028(
R IC0029(
-1
T
RICOO 30{
*G1
G = G1*G2
RIC0031
RIC00321
*****PARAMETER DESCRIPTION:
RIC0033(
ON INPUT:
RIC00341
RICOO35(
CONTAINING
THE
ARRAYS
OF
DIMENSIONS
ROW
NZ,NF,NG,NH
RIC00364
Z (AND W),F,G, AND H, RESPECTIVELY, AS
RIC0037(
DECLARED IN THE CALLING PROGRAM DIMENSION
RIC0038
STATEMENT;
RICO 391
RIC0040(
F,G,H;
MATRICES
THE
OF
ORDER
N
RIC004!
RICi042{
- 2*N = ORDER OF THE INTERNALLY GENERATED
NN
RIC0043!
MATRICES Z AND W;
RICO04 4
RIC0045(
A NONSINGULAR N X N (REAL) MATRIX;
F
RIC001 46~
RIC0047t
N X N SYMMETRIC, NONNEGATIVE DEFINITE
G,H
RICO0048i
(REAL) MATRICES.
RIC0049
R'IC 005O
ON OUTPUT:
RIC0051{
AN N X N ARRAY CONTAINING TH.E UNIQUE POSITIVE RIC052!
H
RIC00534
(OR NONNEGATIVE) DEFINITE SOLUTION OF THE
RIC0054i
RICCATI EQUATION;
RIC0055(
-57-
C
C
C
C
C
C
EREI
RICDSD 2
REAL SCRATCH VECTORS OF LENGTH 2*N; ON OUTPUT RIC00561
(ER (I) ,EI (1)),
I= 1,N
CONTAIN THE REAL AND
RICCO57i
IMAGINARY PARTS, RESPECTIVELY, OF THE N
RIC005W8
CLOSED LOOP EIGENVALUES (I.E., THE
RIC00591
SPECTRUM OF
T
-1
T
RICO060(
F - G* ((G2 + Gl *X*Gt) )*G1 *X*FRIC0061i
C
C
-T
=
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
F -
G*F
*(X
-
RIC0062(
RIC00631
H));
Z,W
2*N X 2*N REAL SCRATCH ARRAYS USED FOR
COMPUTATIONS INVOLVING THE SYMPLECTIC
MATRIX ASSOCIATED WITH THE RICCATI EQUATION;
WORK,SCALE
REAL SCRATCH VECTORS OF LENGTHS N, 2*N,
RESPECTIVELY;
ON OUTPUT, WORK(1) CONTAINS A
CONDITION NUMBER ESTIMATE FOR THE FINAL NTH
ORDER LINEAR MATRIX EQUATION SOLVED;
ITYPE,IPVT
INTEGER SCRATCH VECTORS OF LENGTHS 2*N,
RESPECTIVELY.
***NOTE:
N,
ALL SCRATCH ARRAYS MUST BE DECLARED AND INCLUDED
IN THE CALL.***
*****ALGORITHM NOTES:
IT IS ASSUMED THAT:
(1)
F IS NONSINGULAR
(2)
G AND I ARE NONNEGATIVE DEFINITE
(3)
(F,GI) IS STABILIZABLE AND (C,F) IS DETECTABLE WHERE
T
C *C = H (C OF FULL bANK = RANK(H))
.
RICOO86i
UNDER THESE ASSUMPTIONS THE SOLUTION (RETURNED IN THE ARRAY H) IS RICO087(
UNIQUE AND NONNEGATIVE DEFINITE.
RIC00881
RIC0089(
*****HISTORY:
RfIC0090(
WRITTEN BY ALAN J. LAUB (ELEC. SYS. LAB., M.I.T., RM. 35-331,
RIC0091(
CAMBRIDGE, MA 02139, PH.: (617) - 253-2125), SEPTEMBER 1977.
RICOO92<
MOST RECENT VERSION:
SEP. 15, 1978.
RIC0093i
R IC0094f
C
:-:::':::
C
C
C
C
C
C
EPS IS AN INTERNALLY GENERATED MACHINE DEPENDENT PARAMETER
SPECIFYING THE RELATIVE PRECISION OF FLOATING POINT ARITHMETIC.
FOR EXAMIPLE,
EPS
1=
6.ODO**(-13) FOR DOUBLE PRECISION ARITHMETIC
ON IBM S360/5370.
5
C
C
C
RIC0064I
RIC0065{
RIC0066t
RIC00671
RIC0068S
RIC0069(
BIC0703
IRICOO71(
RIC0072i
RIC00731
RIC007L4
RIC0075S
RIC0076t
RICO0077i
RIC0078{
RIC0079i
RIC080(
RIC0081:
RICO082(
RIC0083(
RIC0084{
RIC0085{
::
.....
:.:**:
EPS= 1. DO
-EiPS=0.5D0*EPS
EPSP1=EPS+ 1 0D0
IF
(EPSPl.GT. 1.ODO)
EPS=2.ODO*EPS
.:
::::.>:::::::::
GO TO 5
SET UP SYMPLECTIC MATRIX Z
DO 20 J=1,N
RIC0096
RIC0097(
RIC0098!
RICOO99t
RICO100I
RICO10 1t
RIC010 2{
RICO0103
RIC104(
RICO01051
RIC0106i
RIC0 107
RICO1083
RICO10 9g
RICO11O1
-58-
DO 10 I=1,N
z (N+I , N+J) =- (J,I)
10
CONTINUE
20
CONTI NUE
CALL DDCOMP (NF, N,F,COND,IPVT, WO K)
CONDP I=COND+ 1. ODO
IF
(C0NDt1. GT.COND) GO TO 30
WRITE (6,44400)
44400 FORMAT (42H1F MATRIX IS SINGULAR TO WORKING
RETURN
30
DO 60 J=1,N
DO 40 I=1,N
WORK {I) =0. ODO
40
CONTINUE
WORK (J)=1.ODO
CALL DSOLVE (NF, N,F, WORK,IPVT)
DO 50 I=1,N
z (I,J)
=WORK (I)
50
CONTINUE
60
CONTINU E
DO 80 J=1,N
DO 70 I=1,N
F (I ,J) =Z (I ,J)
70
CO NTINUE
80
CONTINUE
CALL MULWOA (NHNF,N,H,F,WORK)
DO 120 J-1,N
DO 90 I=1,N
Z (I,N+J)=0. ODO
RICDSD 3
PRECISION)
z (N+I,J)=H (I,J)
90
100
110
120
130
140
C
C
C
CONTINUE
DO 110 K=1,N
DO 100 I=1,N
+F (I,K)*G (K,J)
Z (I,N+J)=Z (t,N+J)
CONTINUE
CONTINUE
CONTINUE
CALL .BLULWOB (NH,NG, N, H, G, WORK)
DO 140 J=l,N
DO 130 I=1,N
Z (N+I, N+J) =Z (N+I, N+J) +G (I,J)
CONTINUE
CONTINUE
BALANCE Z
CALL
C
C
C
C
BALBAANC
(NZ,NN,Z,LOW,IGH,SCALE)
REDUCE Z TO REAL SCHUR FORM WITH EIGENVALUES OUTSIDE THE UNIT
DISK IN THE UPPER LEFT N X N UPPER QUASI-TRIANGULAR BLOCK
NLOW=l
NUP=NN
CALL ORTHES
CALL ORTRAN
(NZ,NN,NLOW,NUP,Z,ER)
(NZ,NN, NLOW,NUP,Z,ER,W)
RICOl111t
RICO1 12i
RICOI1 3
RIC01141
RIC0 1 5
RICO 1 t16
RICO 17i
RICO 1 18(
RICO1019
RIC1 20(
RIC0121!
RIC0322i
RICO123i
RIC01294
RIC01251
RIC01261
RICO 127(
RIC01281
RICO 129(
RI C01 3 01
RIC0131(
RIC01321
RIC01331
PIC0134(
RIC0135{
RIC0 36(
RICO137(
RIC0138(
RICO 139
RI CO140(
RICO0141
RICO 142
RIC0143i
RIC0 1O44
RIC01451
RIC01461
RICO 147(
RIC0!48{1
RIC0149(
RICO1501
RICO 15 1
RIC01 521{
RIC0153(
RIC01541
RIC0155O
RICO156
RICO 1571
RICO 158{
RIC01591
RIC0160(
RICOI6 1
RIC01621
RIC0163i
RIC01641
RICO 165(
-59-
CALL HQR3
C
C
C
C
150
160
C
C
C
RICDSD 4
(Z,,NN,NLOW,NUP,EPS,ER,EI, ITYPE, NZ,NZ)
COMPUTE SOLUTION OF THE RICCATI EQUATION FROM THE ORTHOGONAL
MATRIX NOW IN THE ARRAY W.
STORE THE RESULT IN THE ARRAY H.
CALL BALBAK (NZ,NN,LOW,IGH,SCALE,NN,W)
DO 160 J=1,N
DO 150 I=1,N
F (I, J) W (J, I)
H (I ,J)=W (N+J,I)
CONTINUE
CONTINUE
CALL MLINEQ (NF,NH,N,N,FH,COND,IPVT,WORK)
WORK (1) =COND
TRANSFORM TO GET THE CLOSED LOOP
SPECTRUM
190 I1t,N
IF
(ITYPE(I) .GE.0)
GG TO 170
WRITE (6,44410) I
44410
FORMAT (lX,I4,IX,41HTH EIGENVALUE
RETURN
170
IF
(ITYPE(I).GT. ) GO TO 180
ER (I) = 1. O0D0/ER (I)
EI ({) =0. ODO
GO TO 190
180
IF ({ITYPE(I). EQe.2) GO TO 190
T=ER (1) **2+EI (I) **2
ER (I)= ER (I)/T
EI (I) =EI (I) /T
ER (I+ 1) - ER (I)
EI (I+1) =-EI)
I
190
CONTINUE
RETU RN
C
C
LAST LINE OF RICDSD
C
END
DO
NOT SUCCESSFULLY CALCULATED)
RIC0166
RI C0 1671
RICO168i
RIC01691
RIC0170O
RIC0171i
RIC0172i
RIC0173i
RICO 1741
R C01 75.
RIC0O 176
RIC0177i
RIC0178{
RC0179
RICO 180(
RIC0181I
RICOt821
RIC0183I
RIC0184(
RIC 085(
RIC0186
RIC0187(
RIC01881
RICO 189(
RICO 901
RIC01911
RIC0192(
RIC0193L
RIC 194(
RIC0195i
RICO 196(
RIC01 97
RIC 198(
RIC0 99{
RIC0200(
RIC020 1(
RIC0202{
RICO0203(
-60-
DDCOMP 1
C
DDC000 10
DDC00020
DDC00030
DDC0040
DDC00050
DDC00060
DDC00070
DDC00080
DDC00090
DDC00 100
DDC00 110
DDC00120
DDC00130
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
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C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
DDC00 150
DDCOO160
*****PURPOSE:
N
DDC00170
N
X
THE
REAL
OF
THIS SUBROUTINE CCMPUTES AN LU-DECOMPOSITION
DDC00180
MATRIX A BY GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING.
DDC00190
A
IS
ESTIMATED.
A CONDITION NUMBER OF
DDC00200
DDC O0210
DESCRIPTION:
*****PARAMETER
DDC00220
ON INPUT:
DDC00230
DDC00240
ROW DIMENSION OF THE ARRAY CONTAIN-ING A AS
NA
DDC00250
DiECLARED IN THE CALLING PROGRAM DIMENSION
DDC00260
STATEMENT;
DDC00270
DDC00280
ORDER OF THE MATRIX;
N
DDC00290
DDC00300
N X N MATRIX TO BE TRIANGULARIZED.
A
DDC00310
DDCO 3 20
ON OUTPUT:
DDC00330
DDCO00340
N X N ARRAY CONTAINING AN UPPER TRIANGULAR
A
DDC00350
MATRIX U AND A PERMUTED VERSION OF A LOWER
DDC00360
TRIANGULAR MATRIX I-L SO THAT
DDC00370
(PERMUTATION MATRIX)*A = L*U.
DDC00380
DDC00390
AN ESTIMATE OF THE CONDITION OF A FOR THE
COND
DDC00400
LINEAR SYSTEM
DDC-00410
A*X = B.
DDC00420
CHANGES IN A AND B MAY CAUSE CHANGES COND
IF COND + 1.ODO = COND,DDC00430
TIMES AS LARGE IN X.
DDC00440
COND IS
A IS SINGULAR TO WORKING PRECISION.
DDCOO450
SET TO 1.OD+32 IF "EXACT" SINGULARITY IS
DDC00460
DETECTED.
DDC00470
DDC00480
PIVOT VECTOR CF LENGTH N.
IPVT
D3DC0490
IPVT(K) = THE INDEX OF THE K-TH PIVOT ROW.
DDC00500
IPVT (N) = (-l)**(NUMBER OF INTERCHANGES).
DDCOO510
DDCOO520
REAL SCRATCH VECTOR OF LENGTH N.
WORK
DDC005S30
ITS OUTPUT
ITS INPUT CONTENTS ARE IGNORED.
DDC00540
CONTENTS ARE USUALLY UNIMPORTANT.
DDC00550
SUBROUTINE DDCOMP
C
C
C
C
(NA,N,A,CONL, IPVT,WORK)
*****PA RA ETERS:
INTEGER NA, N,IPVT(N)
DOUBLE PRECISION A(NA,N) COND,WORK(N)
*****LOCAL VARIABLES:
INTEGER NM1,.I,J,K, P1,KB,KM1,M
DOUBLE PRECISION EK,T, ANORM,YNORM,ZNORM
C
**'***FUNCTIONS:
DOUBLE PRECISION DABS
-61-
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C:::::::::
C
C
C
C
5
10
C
C
C
C
C
C
DDCOMP 2
*****APPLICATIONS AND USAGE RESTEICTIONS:
DDC00560
DDCOMP CAN BE USED IN CONJUNCTION WITH DSOLVE TO COMPUTE SOLUTIONSDDC00570
TO SYSTEMS OF LINEAR EQUATIONS. IF NEAEi-SINGULARITY IS
DDC00580
DETECTED SOLUTIONS ARE MORE RELIABLY COMPUTED VIA SINGULAR
DDC0590
VALUE DECOMPOSITION OF A.
DDC00600
DDCOMP CAN ALSO BE USED TO COMPUTE THE DETERMINANT OF A.
DDC00610
ON OUTPUT SIMPLY COMPUTE:
DDCOO620
DET (A) = I PVT (N) *A (1, 1) A (2 ,2)
. . .A
(N,N)
DDC06 30
DDC00640
*****ALGORITHM NOTES:
DDC00650
DDCOMP IS A DOUBLE PRECISION ADAPTATION OF THE SUBROUTINE DECOMP
DDC00660
(SEE REFEERENCE (1) FOR DETAILS).
THIS ALGORITHM IMPLEMENTS
DDC00670
GAUSSIAN ELIMINATION IN A MODERATELY UNCONVENTIONAL MANNER
DDC00680
TO PROVIDE POTENTIAL EFFICIENCY ADVANTAGES UNDER CERTAIN
DDC00690
OPZERATING SYSTEMS (SEE REFERENCE (2) FOR DETAILS).
DDCO0700
DDC007 10
*****jEFERENCES:
DDC00720
(1)
FORSYTHE,G.E.,
MALCOLM,M.A., AND MOLER,C.B.,
COMPUTER
DDC00730
METHODS FOR MATHEMATICAL COMPUTATIONS, PRENTICE-HALL,
1977. DDC00740
(2)
MOLER,C.B.,
MATRIX COMPUTATIONS VITH FORTRAN AND PAGING,
DDC00750
COMM. ACM,
15(1972),
268-270.
DDC00760
DDC00770
*****HISTORY:
DDC00780
ADAPTATION AND DOCUMENTATION WRITTEN BY ALAN J. LAUB
DDC00790
(ELEC. SYS. LAB., M.I.T., RM. 35-331, CAMBRIDGE, MA 02139,
DDC00800
PH.: (617)-253-2125),
AUGUST 1977.
DDC0810
MOST RECENT VERSION:
SEP. 21,
1977.
DDC00820
DDC00 830
:::::::::::::::::::::'::
:::'::.::.:.::::::::DDC0840
DDC00850
IPVT (N) = 1
DDC00860
IF (N.EQ 1) GO TO 80
DDC 00870
N 1t=N- 1
DDC00880
DDC0O890
COHPUTE 1-NORM OF A
DDCO900
DDC0910
ANORM=. ODO
DDCO0920
DO 10 J=1,N
DDC00930
T=. ODO
DDC00940
DO 5 I=1,N
DDC00950
T=T +DABS (a (i,J))
DDC00960
CONTINUE
DDC00970
IF
(T. GT.ANORM) ANOR1=T
DDC00980
CONTINUE
DDC00990
DDC01000
GAUSSIAN ELIMINATION WITH PARTIAL PIVOTING
DDC01010
DDC0 1020
DO 35 K=1,NM
DDC01 030
KP1=K+ 1
DDC010 40
DDC0105
FIND PIVOT
DDC01060
'DDC01070
M=K
DDC01 080
DO 15 I=KP1,N
DDC01090
IF
(DABS(A(I,K)).GT. DABS (A(M,K) )) M=I
DDC01100
-62-
15
C
C
C
CONTINUE
IPVT (K) =M
IF (M. NE.NK) IPVT (N) =-IPVT (N)
T= A (M,K)
A (M,K) =A (K, K)
A (K, K) =T
SKIP STEP IF PIVOT IS ZERO
IF
C
C
C
(T.EQ.0.0D0) GO TO 35
COMPUTE MULTIPLIERS
DO 20
20
C
C
C
25
30
35
C
C
C
C
C
C
C
C
C
C
C
C
40
45
50
I=KP1,N
A (I ,K) =-A (I, K)/T
CONTIN UE
INTERCHANGE
AND ELIMINATE BY COLUMNS
DO 30 J=KP1,N
T=A (M,J)
A (M,J)=A (K,J)
A (K,J)=T
IF (T.EQ.0.ODO) GO TO 30
DO 25 I=KPl,N
A (I J) A (I,J) +A (I, K) *T
CONTIN UE
CON T.fN UE
CONTINUE
COND = (1-NORM OF A)*(AN ESTIMATE OF 1-NORM OF A-INVERSE).
ESTIMATE OBTAINED BY ONE STEP OF INVERSE ITERATION FOR THE
SMALL SINGULAR VECTOE.
THIS INVOLVES SOLVING TWO SYSTEMS
T
WHERE L
AND
A*Z = Y
OF EQUATIONS: A *Y = E
IS A VECTOR OF +1 OR -1 CHOSEN TO CAUSE GROWTH IN Y.
ESTIMATE = (1-NORM OF Z)/(1-NCRM OF Y).
T
SOLVE
A
*Y = E
DO 50 K=1,N
T=0.0DO
IF (K.EQ. ) GO TO 45
KM 1=K-1
DO 40 I=1,KM1
T=T +A (I, K) *WORK (I)
CONTINUE
EK=. 0DO
IF (T..LT. .ODO) EK=- .0ODO
IF {A K,K)..EQ.0.ODO) GO TO 90
WORK (K) =- (EK+T) /A (K,K)
CON TIN U
DO 60 KB=1,NM1
K=N- K B
DDCOMP 3
DDC01110
DDC01120
DDCO1 130
DDCO 1140
DDC01 150
DDCO 1 t60
DDC i 170
DDC01180
DDC01190
DDC01200
DDC01210
D DC 01220
)DDC01230
DDC01240
DDC01250
DDC01260
DDC01270
DDC01280
DDC01290
DDC01300
DDC013 10
DDC0 1320
DDC 1330
DDC01 340
DDC01350
DDC01.360
DDC0 1370
DDCO 138 0
DDC01390
DDC01400
DDC01410
DDCO1420
DDC01430
DDC01440
DDC01450
DDCO1460
DDC01470
DDC0 480
DDC01490
DDC01500
DDC01510
DDC01520
DDC01530
DDC 0540
DDC01 550
DDC01560
DDC0 5 70
DDC01580
DDC01590
DDC01600
DDC01610
DDC 1620
DDC01630
DDC01640
DDC01650
-63-
55
60
C
65
C
C
C
T=O0.00
KP?1K+1
DO 55 I=KP1,N
T=T +A (I, K) *WORK (K)
CONTINUE
WORK (K)=T
M=IPVT (K)
GO TO 60
(M.EQ.K)
IF
T=WORK (M)
WORK (M)= WORK (K)
WORK(K)=T
CONTINUE
YNORM=O.ODO
DO 65 rI = ,N
YNOR M=YNOBR+DABS (WORK (I)
CONTINUE
SOLVE
A*Z
CALL DSOLVE
Y
(NA,N,A,WORK,IPVT)
C
70
C
C
C
ZNOR=0. ODO
DO 70 I=1,N
ZNORM=ZNOih+DABS (WORK (I))
CONTINUE
ESTIMATE
CONDITION
COND=A NORM*ZNORM/YNOR1
IF
(COND.LT.t.ODO) COND=1.ODO
RETURN
C
C
C
80
C
C
C
90
C
C
C
1-BY-1
CASE
COND=1.ODO
IF
(A (1, 1).NE. 0.0 DO)
"EXACT"
SINGULARITY
CoND=1.0D+32
RETURN
LAST
END
LINE OF DDCOMP
RETURN
DDCOMP 4
DDC01 660
DDC01670
DDC0680
DDC01690
DDC01700
DDC017 10
DDC 01720
DDC017 30
DDC01740
DDCO17 50
DDC01760
DDC01770
DDC0 780
DDC01790
DDC01800
DDC0 180
DDC0 1820
DDC01830
DDC 01840
DDC01850
DDC01860
DDC01870
DDC01880
DDC01890
DDC01900
DDC01910
DDC01920
DDC01930
DDC01940
DDC019 50
DDC01960
ODC01970
DDC01980
DDC01990
DDC02000
DDC02010
DDC02020
DDC02030
DDC020L40
DDC02050
DDC02060
DDC02070
DDC02080
DDC02090
DDCO2100
DDC02110
-64-
SUBROUTINE DSOLVE
c
C
C
C
(NA,N,A,B,IPVT)
***** PAAME TER S:
INTEGER NA, N,IPVT(N)
DOUBLE PRECISION A(NA,N) ,B(N)
:@'X***'**LOCAL VARIABLES:
iNTEGER KB,KM1,NM1,KP1,I,K,M
DOUBLE PRECISION T
C
C
C
C
C
C
CC
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
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C
C
C
C
C
C
DSOLVE 1
DS0001(
DSC0002{
DS00003(
DSO000O4(
DSO0005C
DS00006(
DS00007(
DSO0008(
DSO0009(
DS0 t0(
DS00012(
DSO00 13(
DSO0014(
DSC0015(
D3SO0016(
DS000 17(
*c****PARAlifETER DESCRIPTION:
DS(00 18(
ON INPUT:
DSC001 9(
DSO0020(
NA
ROW DIMENSION OF THE ARRAY CONTAINING A
DSO0021(
AS DECLARED IN THE CALLING PROGRAM DIMENSION
DSO0022(
STATEMENT;
DSO0023(
DS00024(
N
ORDER OF THE MATRIX A;
DS00025(
DS 0026
A
TRIANGULARIZED MATRIX OBTAINED FROM DDCOMP;
DSO00027
DS00028(
B
RIGH'T¢' HAND SIDE VECTOR OF LENGTH N;
DS00029(
DS00030(O
IPVT
PIVOT VECTOR OF LENGTH N OBTAINED FROM DDCOMP.DSO0031(
DS00032(
ON OUTPUT:
DS00033(
DS00034(
B
SOLUTION VECTOR, X, OF LENGTH N.
DSO0035(
DS00036C
*****APPLICATIONS AND USAGE RESTRICTIONS:
DS00037(
DSOLVE SHOULD NOT BE USED IN CASE DDCOMP HAS DETECTED NEARDS00038(
SINGULARITY.
SINGULAR VALUE ANALYSIS IS THEN MORE RELIABLE.
DS00039(
DSO0040(
**8***ALGORITHM NOTES:
DS00041(
DSOLVE IS A DOUBLE PRECISION ADAPTATION OF THE SUBROUTINE SOLVE
DS00042(
(SEE REFERENCE (1) IN THE DDCOMP DOCUMENTATION FOR DETAILS).
DSO0043(
DS00-44(
*****HISTORY:
DSO0045(
ADAPTATION AND DCCUMENTATION WRITTEN BY ALAN J. LAUB
DSO0046(
(ELEC. SYS. LAB., M.1.T., RM. 35-331, CAMBRIDGE,
MA 02139,
DS00047(
PH.: (617)-253-2125), AUGUST 1977.
DSO0048
MOST RECENT VERSION:
SEP. 21, 1977.
DSO0049(
DSO00050(
*****PURPOS E
THIS SUBROUTINE SOLVES THE LINEAR SYSTEM
A*X = B
BY FORWARD ELIMINATION AND BACK SUBSTITUTION USING THE
TRIANGULAR FACTORS OF A PROVIDED BY DDCOMP
C
=:=::.:.::,:::::::4::::::::::::::::::::::::::::::::::::::::::::::::::::::::::5
C
C
C
FOR-WARD
IF
ELIMINATION
(N.EQ.1)
GO TO 50
DS00052(
DS00053(
DSO0054(
DS00055(
-65-
10
20
C
C
C
30
40
50
C
C
C
NM 1=N- 1
DO 20 K=1,NM1
KP1=K+1
M-=IPVT (K)
T=B (M)
B(M) = B (K)
B (K) =T
DO 10 I=KP1,N
13 (I)=B (I)+A (I,K) *T
CONTINUE
CONTINUE
BACK
SUBSTITUTION
DO 40 KB=1,NM1
KM1=N-KB
K=KM1+1
B (K) =B(K)/A (K,K)
T=-B (K)
DO 30 I=1,Kfll
B (I) -B (1) +A (I,
CONTINUE
CONTINUE
B (1) =B (1) /A (1,1)
RETURN
LAST
END
LINE OF DSOLVE
K) *T
DSOLVE 2
DS00056(
DS0057(
DSOO0058(
DS00059(
DSO00601
DS00061(
DSO006-2
DS00063(
DSOOO64(
DS00065(
DSO0066(
DSO0067{
DS00068(
DS©00698
DS00701
DSO007t1
DSO0072(
DS00073(
DS00074(
DKS0t75(
DS00076(
DS00077(
DS00078(
Ds1007ts
Ds0080(
DS00O10(
DS00082(
DS00083(
DSO0084(
-66-
S'JBEOUTINE EXCHNG
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
EXCHNG 1
(A,V,N,L, B1,B2,EPS,FAIL,NA,NV)
EXCO0010
EXC000 20
EXCO0030
E XC00040
EXC0050
EXC00060
EXCc0070
EC000oso
EXCOO0090
EXC00100
EXC00 10
EXC0o120
EXC0130
EXCO0 140
EXC0050
EXC0 160
EXC00170
a***** AARAET ER S
INTEGER B1,B2,L,NA,NV
DOUBLE PRECISION A(NA,N),EPS,V(NV,N)
LOGICAL FAIL
*****LOCAL VARIABLES:
INTEGEA I,IT,J,L1,M
DOUBLE PRECISION PQRS,W,X,Y,Z
*****FUNCTIONS:
DOUBLE PRECISION DABS,DSQRT,DMAX1
*****SUBROUTINES CALLED:
QESTEP
e
EXC00190
E XC0200
EXCO0210
EXC00220
EXCO0230
EXC00240
EXCO0250
EXC00260
*****PARAMETER DESCRIPTION:
EXC00270
ON INPUT:
EXCO00280
NA,NV
ROW DIMENSIONS OF THE ARRAYS CONTAINING A
EXC00290
AND V, RESPECTIVELY, AS DECLARED IN THE
EXCO0300
CALLING PROGRAM DIMENSION STATEMENT;
EXC00310
EXCOt.32C
A
N X N MATRIX WHOSE BLOCKS ARE TO BE
EXC0330
INTERCHANGED;
EXC00340
EXC0O350
N
ORDER OF THE MATRIX A;
EXC00360
EXC00370
L
POSITION OF THE BLOCKS;
EXC00380
EXC00390
B1
AN INTEGER CONTAINING THE SIZE OF THE FIRST
EXC00400
BLOCK;
EXCO0410
EXCO042C
B2
AN INTEGER CONTAINING THE SIZE OF THE SECOND
EXC00430
BLOCK;
EXCOO44C
EXC0045C
EPS
A CONVERGENCE CRITERION (CF. HQR3).
EXC00460
EXC0047C
ON OUTPUT:
EXCO0480
EXCO0049C
FAIL
A LOGICAL VARIABLE WHICH IS .FALSE. ON A
EXC00500
NORMAL RETURN. IF THIRTY ITERATIONS WERE
EXC0051C
PERFORMED WITHOUT CONVERGENCE, FAIL IS SET TO EXCO0520
.TRUE. AND THE ELEMENT A(I.+B2,L+B2-1) CANNOT
EXCO053C
Bt ASSUMED ZERO.
EXCOS04C
EXC055C
*****PUR POSE:
GIVEN THE UPPER HESSENBERG MATRIX A WITH CONSECUTIVE Bl X B1 AND
B2 X B2 DIAGONAL BLOCKS (B1, B2.LE.2) STARTING AT A(L,L), THIS
SUBROUTINE
PRODUCES A UNITARY SIMILARITY TRANSFORMATION THAT
EXCHAN;GES THE BLOCKS ALONG WITH THEIR EIGENVALUES. THE
TRANSFORMATION IS ACCUMULATED IN V.
-67-
C
C
C
C
C
C
C
10
20
30
40
50
60
EXCHNG 2
****
HISTOR Y:
DOCUŽMENTED BY J.A.K. CARRIG (ELEC. SYS. LAB., M.I.T., RM. 35-307,
CAMBRIDGE, MA 02139, PH.: (617) - 253-2165, SEPTEMBER 1978.
MOST RECENT VERSION: SEPT. 21, 1978.
:
FAIL= . FALSE.
IF (Bl.'Q.2) GO TO 70
IF
(B2.EQ.2) GO TO 40
L t=L+ 1
Q=A (L+ ,L+1 ) -A (L,L)
P=A (L, L 1)
E=DMAX1 (P,Q)
(R.EQ.0.0
.F
DO) RETURN
P=P/R
Q=Q/R
R=DSQhT (P**2+Q ** 2)
P =P/R
Q=Q/E
DO 10 J=L,N
S=P*A(L,J) +Q*A(L+1,J)
A (L+1,J)=P*A (L+ 1,J)-Q*A(L,J)
A (L,J)=S
CONTINUE
DO 20
=1I,Ll
S-=P*A (, L)+Q*A (I,L+1)
A (I,L+1)=P*A (I,L+l)-Q*A(I,.L)
A (I,L)=S
CONTINUE
DO 30 I=1,N
S=P*V(, L) +Q*V (I,L+1)
v (I ,L+) =P*V (I,.L+1) -Q*V (I,L)
V (I,L) =S
CO NTINJ E
A (L+ 1,L) =0. ODO
RETURN
CONTINUE
X=A (L,L)
P=1. ODO
Q=l1.ODO
-= .ODO
CALL QRSTEP (A,V,P,Q,R,L,L+2,N, NA,NV)
IT =0
IT=IT+1
IF
(IT.LE.30) GO TO 60
FAIL=. TRUE.
RETURN
CONTINUE
P=A (L,L)-X
Q=A (L-+1 ,L)
R=O.ODO
CALL QRSTEP (A,V,P,Q,R,L,L+2,N,NA,NV)
IF (DAt3S(A(L+2,L+1)) .GT.EPS*(DABS(A(L+1,L+1))+DABS(A{L+2,L+2))))
+
GO TO 50
EXC 0056C
EXC00570
EXCOO58C
EXC059C
EXCO060C
XCOO:::::
EXCO062C
EXCOO63
EXC00640
EXC0065C
EXCO0660
EXC0067C
EXCO 680
EXCO0690
EXC00700
EXCO0710
EXCO0720
EXCO0730
EXCOO74O
EXCO0750
EXCO0760
EXCO0770
EXCO0780
EXCO0790
EXCOO800
EXCO0810
EXC0820
EXCOO03C
EXCO840
EXC0085C
EXC00860
EXC00870
EXCO0880
EXC00890
EXC 0900
EXCO09 10
EXCO0920
EXCO0930
EXCOQ940
EXCO0950
EXCO0960
EXCO0970
EXC00980
EXC00990
EXCO 1000
EXC 01010
EXCO 10 20
EXCO1030
EXCO1040
EXCO1050
EXCO1060
EXCO1070
EXC01080
EXC01090
EXCO1t00
-68-
70
80
90
C
C
C
EXCHNG 3
A (Lf 2,L+1) =0.ODO
RETURN
CO NTINUE
M= L+2
IF (B2.EQ.2) Mi=M+1
X=A (L+I,L+1)
Y=A (L,L)
W=A (L+1,L) *A (L,L+l)
P= 1. ODO
Q=1.ODO
R=1.ODO
CALL QRSTEP (A,V,P,Q,R,L,IIN,NA,NV)
I1=0
IT=IT+1
IF (IT.LE.30) GO TO 90
FAIL=. TRUE.
RETURN
CONTINUE
Z=A (L,L)
R=X-Z
S=Y-Z
P= (Rf'*S-W) /A (L+ 1L) +A (L,L+l)
Q=A (L+ 1,L+1)-Z-R-S
R=A (L+2,t
L+)
S=DABS (P) +DABS (Q) +DABS (R)
P=P/S
Q=Q/S
R=P/S
CALL QRSTEP (A,V,P,Q,R,L,M,N,NA,NV)
If (DABS (A (M-1,M-2)) .GT.EPS* (DABS (A (M-1,M-1)) +DABS(A (M-2,M-2)) ) )
+
GO TO 80
A (Pi- 1,M-2) =0.0D0
RETUR N
LAST LINE OF EXCHNG
END
EXCO 1 t10
EXC0 1120
EXCO 1130
E XC0 1140
O
EXCO 150
EXCO1160
EXCO1170
EXC01180
EXCO1290
EXCO1200
EXCO1210
EXCO 220
EXC01230
EXC01240
EXC01250
EXCO 1260
EXCO 1270
EXC01280
EXC01290
EXC01300
EXC01310
EXC0 1320
EXC01330
EXCO1340
EXC 1 3 50
EXC01360
EXC01370
EXCO1380
EXCO1390O
EXC1400
EXC01410
EXC01420
EXC01430
EXC01440
EXC01450
EXCO1460
EXCO0470
-69-
SUBROUTINE HQR3
C
C
C
C
C
C
(A,V,N,NLOW, NUP,EPS,EREI,ITYPE,NA,NV)
*****PARAMETERS:
INTEGER N,NA,NLOW,NUP, NV,ITYPE (N)
DOUBLE PRECISION A(NA,N),EI(N),E (N),EPS,V(NV,N)
*****LOCAL VARIABLES:
LOGICAL FAIL
INTEGER I,IT,L,&U, NL, U
DOUBLE PRECISION Et,E2,P,Q,R,S,T,W,X,Y,Z
*****FUNCTIONS:
DOUBLEZ PRECISION DABS
HQR3 1
HQR0001(
HQR0002(
HQR0003(
HQR0004(
HQROOO05(
HQR0006(
HQR0007C
HQR0008(
HQR0009(
HQROO1 0(
HQRO0 1 1(
HQR 0012(
HQRO0013(
HQR00'14(
HQR0015(
HQR0016(
HQR00 17(
C
C
C
C
***$**SUBROUTINES CALLED:
EXCHNG, QRSTEP, SPLIT
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
HQR001 9
HQRO 20(
*****PURPOSE:
HQeOO21(
A
TO
QUASIMATRIX
HESSENBERG
THIS SUBROUTINE REDUCES THE UPPER
HQR0022(
THE
TRIANGULAR FORM BY UNITARY SIMILARITY TRANSFORMATIONS.
liQROO23(
2
2
X
X
1
AND
THE
1
EIGENVALUES OF A, WHICH ARE CCNTAINED IN
HQR0024(
IN
DESCENDING
ARE
ORDERED
DIAGONAL BLOCKS OF THE REDUCED MATRIX,
HQR0025(
ARE
TRANSFORMATIONS
THE
ORDER OF MiAGNITUDE ALONG THE DIAGONAL.
HQR 0026{
ACCUMULATED .iN THE ARRAY V.
HQ R027(
HQR0028(
DESCRIPTION:
*****PARAMETER
HQR0029(
ON iNPUT:
HQR0030(
A
AND
CONTAINING
ARRAYS
OF
THE
DIMENSIONS
ROW
NA,NV
HQR0031(
THE
CALLING
IN
AS
DECLARED
V, RESPECTIVELY,
H1QR0032{
STATEMENT;
DIMENSION
PROGRAM
HQR0033(
HQR0034(
N X N ARRAY CONTAINING THE UPPER HESSENBERG
A
HQR0035(
BE
REDUCED;
MATRIX TO
HQR0036(
HQR0037(
ORDER OF THE MATRICES A AND V;
N
HQR 0038(
fHQR0039(
A(NLOW,NLOW-1) AND A(NUP,1+NUP) ARE ASSUMED
NLOW,NUP
HQR0040(
TO BE ZERO, AND ONLY ROWS NLOW THROUGH NUP
AND COLUMNS NLOW THROUGH NUP ARE TRANSFORMED, HQR004I1
HQE0042(
RESULTING IN THE CALCULATION OF EIGENVALUES
HQR00431
NLOW THROUGH NUP;
HQRO04(4
A CONVERGENCE CRITERION USED TO DETERMINE WHENHQR0045
EPS
HQR00461
A SUBDIAGONAL ELEMENT OF A IS NEGLIGIBLE.
HQR0047
SPECIFICALLY, A (I+1,i) IS BEGAjRDED AS
HQR0048i
NEGLIGIBLE IF DABS(A (I+1),I)) .LE.EPS*
(DABS (A (I+1,I+1))). THIS MEANS THAT THE FINAL HQ.R00i49
MATRIX RETURNED BY THE PROGRAM WILL BE EXACTLYHQE00503
HQRO05 1(
SIMILAR TO A + E WHERE E IS OF ORDER
FOR ANY REASONABLY BALANCED NORM HQR00521
EPS*NOEM (A),
HQR0053(
SUCH AS THE ROW-SUM NORM;
H QR0054,
HQR0055(
AN INTEGER VECTOR OF LENGTH N WHOSE
I LYPE
-70-
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
I-TH ENTRY IS
0
IF THE I-TH EIGENVALUE IS REAL,
1
IF THE I-TH EIGENVALUE IS COMPLEX WITH
POSITIVE IMAGINARY PART,
2
IF THE I-TI
EIGENVALUE IS COMPLEX WITH
NEGATIVE IMAGINARY PART,
-1 IF THE I-TH EIGENVALUE WAS NOT CALCULATED
SUCCESSFULLY.
HQR 0056
HQR0057i
HQR0058t
HQR00591
HQR0060(
HQR00611
HQR0062(
HQR0063(
HQR0064L
ON OUTPUT:
HQR00651
HQR00661
A
N X N ARRAY CCNTAINING THE REDUCED, QUASIHQR0067i
TRIANGULAR MATRIX;
HQR0068i
HQR0069(
V
N X N ARRAY CCNTAINING THE REDUCING
HQR0070I
TRANSFORMATIONS TO BE MULTIPLIED;
HQR0071(
HQR0072{
ER, EI
REAL SCRATCH VECTORS OF LENGTH N WHICH ON
HQE0073(
RETURN CONTAIN THE REAL AND IMAGINARY PARTS,
HQE0074(
RESPECTIVELY, OF THE EIGENVALUES.
HQR0075{
HQR0076i
*****HISTORY:
HQR0O77(
DOCUbENTED BY J.A.K. CARRIG, (ELEC. SYS. LAB., M.I.T., RN. 35-307,HQR0078(
CAfBRIDGE, MA 02139, PH.: (617)- 253-2165), SEPT 1978.
HQR0079t
MOST RECENT VERSION: SEPT 21,
1978.
HQR0080(
HQR0081(
: ... ::::::
:::::::::::::::::::::::::::
:::Q082(
HQR0083(
DO 10 I=NLOW,NUP
HQR0084(
ITYP
10
20
30
40
50
60
HQR3 2
(I)
=- 1
CONTINUE
T=0.ODO
NU=NU P
IF
(NU.LT.NLOW)
GO TO 240
IT=O
CONTINUE
L=NU
CONTINUE
IF
(L.EQ.NLOW)
GO TO 50
IF
(DABS (A (L,L-1 ) . LT. EPS* (DABS (A (L- 1,L-) ) +DABS (A (L,L)
+
GO TO 50
L=L- 1
GO TO 40
CONTINUE
X=A (NU, NU)
IF (L. EQ. NU) GO TO 160
Y=A (NU- 1 ,NU-1)
W=A(NU, NU-1 ) *A (NU-1.NU)
IF (L.EQ.NU-1) GO TO 100
IF (IT. EQ.30) GO TO 240
IF (IT.NE.10 .AND. IT.NE.20) GO TC 70
T=T+X
DO 60 I=NLOW,NU
A (I I) =A (I,I) -X
CONTINUE
HQR085(
HQERO86(
HQR0087(
HQR0088(
H QR0089(
HQR0090i
HQR009 1(
HQR0092t
HQR0-093(
IIQR0094
))
HQR0095(
HQR0096(
HQR 0097(
HQR0098(
HQR0099{
HQR0100(
HQR01 01(
HQRO 10 2(
HQRO1 03(
HQROo104(
HQR01O5(
HQRO106(
HQRO 107(
HQRO108(
HQRO 1 0 9
HQR0110(
-71-
70
80
90
100
110
S=DABS (A (NU,NU-1)) +DABS(A (NU-1, NU-2))
X=0. 75DOS
Y=X
W=-O. 4375DO*S**2
CONTINUE
HQ.R01111t
HQR01 12(
HQRO1 13(
HQR01 14
HQR01 15(
IT=IT+I
HQR01161
NL=NU-2
CONTINUE
Z=A (NL, NL)
R=X-Z
S=Y-Z
P=(R*S-W)/ A(NL+1,NL) +A(NL,NL+1)
S
Q=A (NL+1 NL+ 1 ) -Z -R=A, { NL+ 2, NL+1)
1
S-DABS (P)+DABS (Q)+DABS (R)
P=P/S
Q=Q/S
R=R/S
IF
(NL.EQ.L) GO TO 90
IF (DABS (A(NL,NL-1)) *(DABS(Q) +DABS(R)).LE.EPS*DABS(P)*
+
(DABS (A (NL-1,NL-1))+DABS () +DABS (A (NL+1 ,NL+1)))) GO TO 90
LNLL-1
GO TO 80
CONTINUE1
CALL QRSTEP (A,V,P,Q,R,NL,NUN,NA,NV)
GO TO 30
IF (NU.NE.NLOW+1) A(NU- 1,NU-2)=0.ODO
A (NU,NU) =A (NU, NU) +T
A(NU-1, NU- ) -A (NU-1,NU-1) +T
ITYPE (NU)=
ITYPE (NU-1) =
MU=NU
CONTINUE
NL=MU-1
CALL SPLIT (A,V,N,NL, EL,E2,NA,NV)
IF (A(MU,MU-1).EQ. 0.ODO) GO TC 170
IF (14U.EQ.NUP) GO TO 230
IF (MU.EQ.NUP-1) GO TO 130
IF (A(MU+2, MU+1) .EQ.0.0D0) GO TO 130
.GE.A(1U+1,AiU+1)*
UIF (A(U-1,MU-1)*A(MU,MU)-A(MiU-1,MU)*A(MU,MU-1)
+
A(aU+2,MU+2)-A(LIU+1,MU+2)*A(MU+2,MU+1)) GO TO 230
CALL EXCHNG (A,V,N,NL,2,2,EPS,FAIL,NA,NV)
IF (.NOT.FAIL) GO TO 120
ITYPE (NL) =- i
120
130
HQR3 3
ITYPE (NL+1) =-1
ITYPE (NL+ 2) =- 1
ITYPE (NL+3) =-1
GO TO 240
CONTINUE
MU=MU+2
GO TO 150
CONTINUE
IF
(A (MU-1,MU- 1) *A (MU,MU) -A (U- 1,MU) *A (MU, TU-1) .GEo
+
A( U+1,MU+1)**2) GO TO 230
CALL EXCHNG (A,V,N,NL,2,1,EPS,FAIL,NA,NV)
HQR0117(
HQ.l01 18{
HQEi0119{
HQ 0120(
HQRO 121(
HQR0122(
HQR0123
HQ0 24(
HQE0125(
HQR0 126(
HQR0127t
HQ0R 128(
HQR0 129(
HQRE0130
HQReO131(
HQRe0132f
HQR0133(
HQR01 34(
HQr0135(
HQR0136(
HQ.1O 1 37(
HQRO 1 38(
aQRE01391
HQR0140(
HQ R 0141(
HQR0142(
HQF 014 3(
HQR0144(
HQR0145(
HQR0146(
HQR0147(
HIQRO1481
HQR0149i
HQR0150(
HQR0151(
HQBO0152{
HQ.RO 153(
HfQR0154(
HQR0 1 55(
QR0 156(
HQOR01571
HQR0158(
HQR01 59(
HQR 160(
HQE0161(
HQR0162(
HQR0163(
HQR01644(
HQR0165(
-72-
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
HQR3 4
IF (.NOT.FAIL) GO TO 140
ITYPE (NL) =- 1
ITYPE (NL+1) -ITYPE (NL+2) -1
GO TO 240
CONTINUE
MU=MU+ 1
CONTINUE
GO TO 110
NL=O
A (NU, NU) =A (NU,NU) +T
IF
(NU. NE.NLOW) A (NU,NU-1)=0. ODO
ITYPE (NU)=0
tU=NU
CONTINUE
CONTINUE
IF
(MU. EQ NUP) GO TO 220
IF (MU.EQi.NUP-1) GO TO 200
IF (A (U+2,MU+1).EQ.0.ODO) GO TO 200
IF (A (NU,MU) **2.GE.A(MU+1,MU+1)*A (pU+2,MU+2)-A(MU+1,MU+2)*
+
A(MU+2,MU+1))
GO TO 230
CALL EXCHNG (A,V,N,MU,1,2,EPS,FAIL,NA,NV)
IF (.NOT.FAIL) GO TO 190
ITYPE (MU) =- I
ITYPE (MU+1) -1
ITYPE (MU+2) =GO TO 240
CONTINUE
MU=MU+2
GO TO 210
CONTINUE
DABS (A (MU+1,MU+1))
GO TO 220
IF (DABS (A (MU,MU)) .GE.
CALL EXCHNG (A,V,N,MU,1,1,EPS,FAIL,NA,NV)
MU=U+ 1
CONTINUE
GO TO 180
CONTINUE
MU=NL
NL=0
IF (MU.NB.0) GO TO 170
CONTINUE
NU=L-1
GO TO 20
IF (NU.LT. NLOW) GO TO 260
DO 250 I=1, NU
A (I,I)=A {I, I)+T
CO TINUE
CONTINUE
N U=NUP
CONTINUE
IF (ITYPE (NU).NE.-1) GO TO 280
NU=NU- 1
GO TO 310
CONTINUE
IF (NU.EQ.N.LOW) GO TO 290
QR 01 66(
HQ0 167(
HQa0168(
HQR0169(
HQR0170(
HQR0171{
HQRO 17 2(
HQIE0173(
HQR0174(
HQR0175(
HQR0176(
HQR0 177(
fHQR0178
HQR0 179(
HQR0180(
HQR0181(
HQE1t82(
HQE0183(
HQE0184(
HQP0185(
HQiR0186(
HQ;R0187{
HQR 188(
HQ1O 891
HQR0 190(
HQi 019 1
HQE0192(
HQR0193(
HQR0194(
HQR 0195(
H'QE0196(
fHQE0197(
HQRE0198(
HQR0199(
HQ.RO200(
fIQR0201(
HQR0202(
HQ0 20 3(
HQR0204(
HQRO 205(
HQR0206(
HQR 020 7(
HQE0208(
HQR0209(
HQR0210(
HQR0211(
HQEO212(
HQ 0213 (
HQ0214(
HQ
2 1 5(
HQE3021 6
HQR0217(
HQP,0218(
HQBR0219(
HQE0220(
-73-
290
300
310
IF (A(NU, NU-1).EQ.0.ODO) GO TO 290
CALL SPLIT (A,V,N, NU-1,El,E2,NA,NV)
IF (A (NU,NU-1).LQ.0.O)D0) GO TC 290
EL (NU)=Et
EI (NU- 1) =E2
ER (NU-1) =ER (NU)
EI (NU)=-- I (NU-1)
ITYPE (NUI -) =1
ITYPE (NU)=2
NU=NU-2
GO TO 300
CONTINUE
ER(NU)=A (NU,NU)
El (NU)=0 ODO
NU=NU-1
CONTINUE
CONTINUE
IF
(NU.GE.N.LOW) GO TO 270
RETURN
.I
C
C
LAST LINS OF HQR3
C
END
HJR3 5
HQP02211
HQii0222(
HQa 022.3
HQE0224(
HQR0225{
HQEf0226(
HiQR 0227(
HQR0228{
HQR0229(
HQR02301
HQR0231(
HQR0232{
HQR0233(
HQR0234(
HQR0235(
HQR0236(
HQR0237(
HQa0238(
HQR0239{
HHQR0240(
HQR 0241t
fIHQ0242(
HQR0243(
HQR0244(
-74-
SUBROUTINE
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
MLINEQ
MLINEQ 1
(NA,NB,N,M,A,B,COND,IPVT, WOEK)
*****PARAMETERS:
INTEGER NA, NB, N,M, IPVT N)
DOUBLE
PRECISION A(NA,N),B(NB,M),COND,WORK(N)
*****LOCAL VARIABLES:
INTEGER I,J,KIN, FOUT
DOUBLE PRECISION CONDP1
*****SUBLOUTINES CALLED:
DDCOMP,DSOLVE
ML0001(
MLI0002(
MLt10003(
MLI 000 4(
MLIS005(
ML10006(
MLIO 007(
MLI0008(
ML10009(
MLI001 0(
MLI0011(
MLI0012(
MLI0013(
MLI0015(
MLI0016 (
MLIO017(
MLI 0018
MLI0019(
MLI0020(
MLIO0021(
MLTO0022(
MLI0023(
MLI0024(
MLI0025(
MLI0026(
MLI0027(
* ***PARAdiETER DESCRIPTION:
NMLI0028(
ON INPUT:
MLI0029C
ML10030(
NA,NB
ROW DIMENSIONS OF THE ARRAYS CONTAINING A AND M.LI0031(
B, RESPECTIVELY, AS DECLARED IN THE CALLING
MLI0032(
PROGRAM DIMENSION STATEMENT;
MLI0033(
MLI0034t
N
ORDER OF THE MATRIX A AND NUMBER OF ROWS OF
MLI0035(
THE MATRIX B;
MLI0036(
MILO0037(
A
NUMBER OF COLUMNS OF THE MATRIX B;
ML10038(
ML10039(
A
N X N COEFFICIENT MATRIX;
MLI0040(
MLI0041 C
B
N X M RIGHT HAND SIDE MATRIX.
MLI0042(
MLI0043C
ON OUTPUT:
MLI0044(
MLI0045(
B
SOLUTION MATRIX
X = (A-INVERSE)*B;
MLI0046(
MLI0047(
COND
AN ESTIMATE OF THE CONDITION OF A;
MLI0048C
ML10049(
IPYT
PIVOT VECTOR CF LENGTH N (SEE DDCOMP
MLTI050C
DOCUMENTATION);
MLI0051(
MLI0052C
WORK
A REAL SCRATCH VECTOR OF LENGTH N.
MLI0053C
ML100 54(
*****A.PPLICATIONS AND USAGE RESTRICTIONS:
MLI0055C
*****PURPOSE :
THIS SUBROUTINE SOLVES THE MATRIX LINEAR EQUATION
A*X = B
WHERE
A IS AN N X N (INVERTIBLE) MATRIX AND B IS AN N X M
MATRIX.
SUBROUTINE DDCOMP IS CALLED ONCE FOR THEI LU-DECOMPOSITION OF A AND SUBROUTINE DSOLVE IS CALLED M TIMES FOR
FORWARD ELIMINATION AND BACK SUBSTITUTION TO PRODUCE THE
M COLUMNS OF THE SOLUTION MATRIX X = (A-INVERSE) *B.
AN
ESTIMATE OF THE CONDITION OF A IS RETURNED.
SHOULD A BE
SINGULAR TO WORKING ACCURACY, A MESSAGE TO THAT EFFECT IS
PRODUCED.
-75-
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
MLINEQ 2
(1)THE VALUE OF COND SHOULD ALWAYS BE CHECKED BY THE CALLING
PROGRAM.
SHOULD A BE NLAR-SINGULAR (OR SINGULAR TO WORKING
ACCURACY) THE DATA SHOULD BE INVESTIGATED FOR POSSIBLE
EREORS.
IF THERE ARE NONE AND THE PROBLEM IS APPARENTLY
WELL-POSED AND/OR MEANINGFUL,
SINGULAR VALUE ANALYSIS IS
THEN A MORE RELIABLE SOLUTION TECHNIQUE (CF. EISPACK
SUBROUTINES
SVD
AND
MINFIT).
(2)MLINEQ CAN BE USED TO COMPUTE THE INVERSE OF A:
SIMPLY SOLVE
A*X = I WHERE
I
iS THE N X N IDENTITY MATRIX.
(3)IF THE SOLUTION TO X*A = B
(X = B*(A-INVERSE)) IS DESIRED,
SIMPLY TRANSPOSE THE SOLUTION CF
T
T
A* X = B
*****ALGORITHM NOTES:
THE CONTENTS OF A ARE MODIFIED BY THIS SUBROUTINE.
SHOULD THE
ORIGINAL COEFFICIENTS OF A BE NEEDED SUBSEQUENTLY, THE
CONTENTS OF A SHOULD BE SAVED PRIOR TO THE CALL TO MLINEQ.
***** HISTORY:
WRITTEN BY ALAN J. LAUB
(ELEC. SYS. LAB., M.I.T., RM. 35-331,
CAMBRIDGE,
MA 02139,
PH.: (617)-253-2125),
AUGUST 1977.
MOST RECENT VERSION:
SEP. 21, 1977.
Ž1LI0056(
ML10057(
MLI0058(
MLI059(
MLIOO60(
MLI0061(
ML10062(
ML10063(
MLI0064(
MLI0065(
ML10066(
MLI0067(
ML 0068(
MLI0069(
ML10070C
MLI0071(
MLI0072(
MLII0073(
MLI007 4(
MLI0075(
MLI0076(
MLI0077C
MLI0078(
MLI0079(
C
MtiLI008 1(
COMMON/INOU/KIN, KOUT
CALL DDCOMP (NA,N,A,COND,IPVT,WORK)
CONDP 1=COND+1. ODO
IF (CONDPl.GT.COND) GO TO 100
WRITE (KOUT,44400)
44 4 00 FORMAT (40HIMATRIX IS SINGULAR TO WORKING PRECISION)
RETURN
100
DO 400 J=1,M
DO 200 I=1,N
WORK (I)=B (I,J)
200
CONTINUE
C
C
COMPUTI
(JU-TH COLUMN OF X) = (A-INVERSE)*(J-TH COLUMN OF B)
C
CALL DSOLVE (NA,N,A,WORK,IPVT)
DO 300 I=1,N
B (I, J) =WORK (I)
300
CCNTINUE
400
CONTINUE
RETURN
C
C
LAST LINE OF MLINEQ
C
END
MLI0082(
ML1008:3(
MLI0084(
MLI:0085(
MLI0086C
ML10087(
MLI0088C
M1LI0089(
MLI0090C
MLI0091(
MLI0092(
MLI 0093(
MLI009;4(
MLI0095(
MLI 0096(
MLI0097(
MLI0098(
MLI0099£
MLIO100(
MLIO 101
MLI0102
£
MLI0103(
MLI0104(
ML10105(
-76-
MUL00010
¶MUL00020
MUL00030
*****PARAMETERS:
MUL00040
INTEGER NA, NB, N
MUL00050
DOUBLE PRECISION A(NA,N),B (NB,N) ,WO8RK(N)
MULOO00060
MUL00070
*****LOCAL VARIABLES:
MUL0080
INTEGER I,J,K
MUL00090O
MUL001 00
*****SUBROUTINES CALLED:
MULOo 110
NONE
MUL00120
:ULOI0
................
aMUL00140
MUL 00150
*****PURPOS :
MUL00160
PRODUCT
MATRIX
WITH
THE
A
THE
ARRAY
THIS SUBROUTINE OVERWRITES
MUL00170
AND B ARE N X N ARRAYS AND MUST BE DISTINCT.
BOTH A A
A*B.
MUL00180
MUL00190
*****PARAMETER DESCRIPTION:
MUL00200
ON INPUT:
MUL00210
AND
MUL00220
A
CONTAINING
ARRAYS
THE
OF
ROW DIMENSIONS
NA,NB
MUL00230
CALLING
THE
IN
AS
DECLARED
RESPECTIVELY,
B,
MUL00240
STATEMENT;
DIMENSION
PROGRAM
iUL00250
MUL00260
B;
AND
A
MATRICES
OF
THE
ORDER
N
MUL00270
280
MUL00
MATRIX;
X
N
AN
N
A
MU.L00290
MUL00300
AN N X N MATRIX.
B
MUL00310
MUL00320
ON OUTPUT:
MUL00330
MUL00340
AN N X N ARRAY CONTAINING A*B;
A
MUL00350
MUL00360O
A REAL SCRATCH VECTOR OF LENGTH N.
WORK
MUL00370
MUL00380
*****HISTORY:
MUL00390
LAUB (ELEC. SYS. LAB., M.I.T., RM. 35-331,
WRITTEN BY ALAN J.
MULO0400
1977.
SEPTEMBER
(617)-253-2125),
PH.:
MA
02139,
,
CAMBRIDGE
MUL00410
MOST RECENT VERSION: SEP. 21, 1977.
MUL00420
========
=
::=
=
===============
::::::::
=::::::::::::::::::
MUL00440
MUL00450
DO 40 I=1,N
MUL00460
DO 20 J=1,N
MUL0470
3 =0. ODO
WOR K (J)
MUL00480
DO 10 K=1,N
MUL00490
WORK (J) =WORK (J) +A (I ,K) *B (K,J)
MUL00500
CONTINUE
MUL 0510
CONTINUE
MUL00520
DO 30 J=t,N
MUL00530
A (I,J)=WORK (J)
MUL00540
CONTINUE
MUL00550
CONTINUE
SUBROUTINE
C
C;
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
-c
C
10
20
30
40
MULWOA 1-
MULW OA
(NA, NB,N,A,B,WB'CK)
30
-77-
RETURN
C
C
C
LAST LINE OF MULWOA
END
MULWOA
2
UL00560
iUL00570
.MULO0580
iM1UL00590
MULO0600
a
-78-
SUBROUJTINE MULWOB
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
(NA,NB,N,A,B,WORK)
*****PARAMETERS:
INTEGER NA,NB,N
DOUBLE PRECISION A (NA,N) ,B(NB,N)WORK (N)
*****LOCAL VARIABLES:
INTEGER I,J,K
*****SUBROUTINES CALLED:
NONE
20
30
MUL000 10
MUL0002O
MU LOO3 30
MUL00040
MUL00050
MUL00060
MUL00070
MUL000 80
MUL00090
MUL00100
MUL001O
C
MULO0120
MUL00140
M'JLO 150
MUL00160
MUtL0017C
MUL00130
*****P ARAMETER DESCRIPTION:
MUL00190
ON INPUT:
MUL0020C
MUL00210
NA,NB
ROW DIMENSIONS OF THE ARRAYS CONTAINING A AND MUL0022C
B, RESPECTIVELY, AS DECLARED IN THE CALLING
MUL0230
PROGRAM DIMENSION STATEMENT;
NULOO24C
MUL0025C
N
ORDER OF THE MATRICES A AND B;
MUL0026C
MU L0027C
A
AN N X N MATRIX;
MUL0028C
¶U1002 9(
B
AN N X N MATRIX.
MULO030C
MUL0031 (
ON OUTPUT:
MUL0032C
MULO033C
B
AN N X N ARRAY CONTAINING A*B;
MULO0.34C
MUIL0035C
WORK
A REAL SCRATCH VECTOR OF LENGTH N.
MULOO36(
MULJ0037C
*****HISTORY:
MUL0038(
WRITTEN BY ALAN J.
LAUB (ELEC. SYS. LAB., M.I.T., RM. 35-331,
MUL0039C
CAMBRIDGE,
MA 02139,
PH.: (617)-253-2125),
SEPTEMBER 1977
MULt0040(
MOST RECENT VERSION: SEP. 21, 1977.
M3UL0041C
*,***PURPOSE:
THIS SUBROUTINE OVERWRITES THE ARRAY 3 WITH THE MATRIX PRODUCT
A*B.
BOTH A AND B ARE N X N ARRAYS AND MUST BE DISTINCT.
C
10
MULWOB 1
DO 50 J1,N
DO 10 I=1,N
WOR K (I)= O. ODO
CONTINUE
DO 30 K=1,N
DO 20 1=1,N
WORK (I) WORK (I) +A(I, K)*B(K,J)
CONTINUE
CONTIN UE
DO 40 I=1,N
B (I ,J) =WZORK (I)
MUL0044(
MUL0045(
MULO '46(
MUUL0047(
MULO048(
MUL0049(
MUL0050(C
MUL0051(
MUL0052C
MULOO53(
MUL0054(
tMUL0055(
-79-
40
50
C
C
CONTINUE
CONTINUE
RETURN
MULWOB 2
MUL0056C
MUL00570
LAST LINE OF MULWOB
MUL 058C
tI[UL0059C
MiUL0600
CND
MUL00620
-80-
SUB3OUTINE QESTE:P
C
c
C
C
C
C
(A,V,P,Q,R, N,
*****PARAMETERS:
INTEGER NNANL,NU,NV
DOUBLE PRECISION A(NA, N)
NU, NNANV)
P,Q,R,V(NV,N)
*****LOCAL VARIABLES:
LOGICAL LAST
INTEGER i,J,K,NL2,NL3,NUM1
DOUBLE PRECISION S,X,Y,Z
*****FUNCTIONS:
DOUBLE PRECISION DABS,DSQRT
QRSTEP 1
QRS000 1C
QRS0OO2(
QRS0003(
QRSO004(
QRS0005C
QRS00064
QRS0007(
QRSOO08(
QRSOOO9(
QRS0010{
QRS0011(i
QRS0 12(
QRS0013i
QR SOO 14
QRS0015t
QRS0 16(
QRS0017{
C
*****SUBROUTINES CALLED:
C
C
NONE
C
C
:::::....::.............QRS00l1
QBS 0 1 91
C
QRS0020!
C
*****PURPOSE:
Q S02 1
C
THilS SUBROUTINE PERFORMS ONE IMPLICIT QR STEP ON THE UPPER
QRS0022!
C
HESSENBERG MATRIX A. THE SHIFT IS DETERMINED BY THE NUMBERS P,Q,
AND R, AND THE STEP IS APPLIED TC ROWS AND COLUMNS NL THROUTH NU. QRS00231
C
QRS0O024
THE TRANSFORMATIONS ARE ACCUMULATED IN THE ARRAY V.
C
C
QRS 0025
QRS00261
C
*****PARAMETER DESCRIPTION:
QRS0027
ON INPUT:
C
QRSOO28
ROW DIMENSIONS OF THE ARRAYS CONTAlNING A
NA,NV
C
QRS0029
AND V, RESPECTIVELY, AS DECLARED IN THE
C
C
CALLING PROGRAM DIMENSION STATEMENT;
QRS0030:
QRS003 1
C
N X N UPPER HESSENBERG MATRIX ON WHICH THE QR QRS0032
C
A
QRS0033 i
STEP IS TO BE PERFORMED;
C
QRS0034
C
QRS0035
PARAMETERS WHICH DETERMINE THE SHIFT;
P,QR
C
C
QRS0036
QRS0037
THE LOWER LIMIT OF THE STEP;
NL
C
QRSO 38
C
QRS0039
THE UPPER LIMIT OF THE STEP;
C
NU
QRSQ0040
C
QRSOO41
ORDER OF THE MATRIX A.
N
C
QRS0042
C
Q S50043
ON OUTPUT:
C
QRS0044
C
QRS0045
N X N REAL SCRATCH ARRAY CONTAINING THE
V
C
QRS0046
ACCUMULATED TRANSFORMATIONS.
C
C
QRS0047
QBS 0048
C
*****HISTORY:
DOCUMENTED BY J.A.K. CARRIG (ELEC. SYS. LAB., M.I.T., RM. 35-307, QRS0049
C
QRS050
C
CAMBRIDGE, MA 02139, PH.: (617) - 253-2165), SEPTEMBER 1978.
QRS0051
MOST RECENT VERSION: SEPT 21, 1978.
C
QRS 0052
C
:
::QRS0053
-5 3:-:
)3:
:: :1:
.
:::::-:-·
:
C...QRS3::'
C
QRS0054
NL2=NL+2
QRS0055
-81-
10
20
30
40
50
60
70
80
90
DO 10 I=NL2,NU
A (1,I-2) =0. D
CONTINUE
IF (NL2,.EQ. NU) GO TO 30
NL3=NL+3
DO 20 I=NL3,NU
A (I,t-3) =0. DO
CONTINUE
CONTINUE
NU&1=NU-1
DO 130 K=NL,NUM1
LAST=K. EQ. NUM1
IF
(K.EQ.NL) GO TO 40
P=A (K,K-1)
Q-A (K+ 1,K-1)
R=0.ODO
IF (.NOT.LAST) fR=A(K+2,K-1)
X=DABS (P) +DABS (Q) +DABS (R)
IF (X.EQ.0. ODO) GO TO 130
P=P/X
Q=Q/X
R=R/X
CONTINUE
S=DSQRT (P**2+Q**2+R**2)
IF
(P.LT.O.ODO) S=-S
IF
(K.EQ.NL) GO TO 50
A (KK-1)=-S*X
GO TO 60
CONTINUE
IF (NL.NE.1) A
-A(K,K--1
CONTINUE
P=P+S
QRSTEP 2
QRS00561
QRS00571
QRS0058(
QRS0059i
QRS0060j
QRS061
QRS0062 ~
QR SO 0631
QRSOO64
QRS0065(
QR50066:
QR S0067t
QRS00681
QRS00691
QRS007 0j
QRS00711
QRS007 2
QRS0073
QRS0074(
QRS0075i
QRS0076i
QR50077i
QRS0078i
QRS0079i
QRS008Oi
QRS00811
QRSO082i
QRS0083(
QRS00841
QRS00851
QRS0086i
QeRSO 87
X=?/S
QRSO088i
Y=Q/S
QRS00891
Z=R/S
QRS0090(
Q=Q/p
R=a/P
DO 80 J=K,N
P=A (K,J) +Q*A (K +1 ,J)
IF (LAST) GO TO 70
P=P+R*A (K+2,J)
A (K+2,J)=A (K+2,J) -P*Z
CONTINUE
A (K+ 1 ,J) =A (K+1 ,J) -PY
A (K,J)=A (K,J)-P*X
CONTINUE
J=MIN0 (K+3, NU)
DO 100 I=1,J
P=X*A (I,K) +Y*A (I,K+1)
IF (LAST) GO TO 90
P=P +Z*A (I, K+2)
A (I ,K+2) =A (I,K+2) -P*E
CONTINUE
A (I,K+ 1) =A (I,K+1)-P*Q
QRSOO91QiRS0092
QRS0093t
QRS00944
QRS0095(
QiS 0O961
QRS0097(
QRS00981
QRS0099
QRS0100i
QRS0101I
QR SO 1021
QRS0103,
QRS0104L
QRS0105i
QR S 106(
QSI01 073
QRS0108,
QRS01 09
A (I ,K)=A (I,K) -P
QRsO
1t0
-82-
100
110
120
130
C
C
C
CONTI NUE
DO 120 I=1,N
P=-XY II, K) +Y*V (I, K+1)
IF (LAST) GO TO 110
P=P+Z*V (I, K+2)
V (IK+2)
=V (I,K+2)-P*R
CONTINUE
V (I,K+ 1) =V (I,K+1) -P*Q
v (I, K) V (I,K) -P
CONTINUE
CONTINUE
RETURN
LAST LINE OF QRSTEP
END
QRSTEP 3
Q R SO 1 1 1
QERS012
QRSO01 13
QRS01 14i
QRSO 15t
QRS0116t
QRSO 17i
QS01 18i
QRS01 19P
QRS0120i
QRS0121,
QRS0 122
QRS0123
QRS50124
QRSO125t
QRS01261
-83-
SUBROUTINE
C
C
C
C
C
C
SPLIT
(A,V,N,L,E1,E2,NA,NV)
*****PARAMETERS:
INTEGER L,N,NA,NV
DOUBLE PRECISION A (NA,N),V(NV,N),E1,E2
***'**LOCAL VARIABLES:
INTEGER I,J,Lt
DOUBLE PRECISION PQRTUWXY,Z
****FUNCTIONS:
DOUBLE PRECISION DAiBS,DSQRT
SPLIT 1
SPLOO 10
SPL00020
S PLOOO30
SPL00040
SPL00050
SPL00060
SPL00070
SPL00080
SPL00090
SPLO100l
SPL001 10
SPLOO120
SPL00130
SPL00O1-40
SPL00150
SPL00 160
:: S:::::
-::
:SP0170
C
C
C
C
:C..
****SUBRO:UTINES CALLED:
NONE
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SPL00 180
SPLOO190
*****PURPOSE:
GIVEN TiiE UPPLR-HESSENBERG MATRIX A WITH A 2 X 2 BLOCK STARTING ATSPLOO200
SPL00210
A (L,L), THIS PROGRAM DETERMINES IF THE CORRESPONDING EIGENVALUES
SPL00220
ARLE EAL OR COMPLEX. IF THEY ARE REAL, A ROTATION IS DETERMINED
SPL00230
THE
THAT REDUCES THE BLOCK TO UPPER-TRIANGULAR FORM WITH
SP.L00240
THE
ELIGENVALUE OF LARGEST ABSOLUTE VALUE APPEARING FIRST.
SPL00250
ROTATION IS ACCUMULATED IN THE ARRAY V.
SPL00260
SPLOO 270
*****PARAMETER DESCRIPTION:
SPLOO28C
ON INPUT:
SPL00290
ROW DIMENSIONS OF THE ARRAYS CONTAINING
NA,NV
SPLOO300
IN
THE
DECLARED
AS
V,
RESPECTIVELY,
A AND
SPLOO310
CALLING PROGRAM DIMENSION STATEMENT;
SPL0032C
THE UPPER HESSENBERG MATRIX WHOSE 2 X 2 BLOCK SPL00330
A
SPL00340
IS TO BE SPLIT;
SPLOO35C
SPL00360
A;
MATRIX
OF
THE
ORDER
N
SP L0037C
SPLOO380
BLOCK.
2
X
2
THE
OF
POSITION
L
SPLOO39C
SPLOO40C
ON OUTPUT:
SPL00O41C
SPL0042C
AN N X N ARRAY CONTAINING THE ACCUMULATED
V
SPL00430
SPLITTING TRANSFORMATION;
SPL400L4C
REAL SCALARS. IF THE EIGENVALUES ARE COMPLEX, SPLO045C
Et,L2
SPL00O46
El AND E2 CONTAIN THEIR COM21MON REAL PART AND
SPL00347C
POSITIVE IMAGINARY PART (RESPECTIVELY).
SPL0048(
IF THE EIGENVALUES ARE REAL, El CONTAINS THE
ONE LARGEST IN ABSOLUTE VALUE AND E2 CONTAINS SPL0049C
SPL00O50C
THE OTHER ONE.
SPLO051C
SPLOO52C
*****HISTORY:
SPL0053(
35-307,
DOCUMENTED BY J.A.K. CARRIG (ELEC. SYS. LAB., M.I.T., E.
SPL005C
SEPT 1978.
MA 02139, PH.: (617) - 253-2165),
CAMBRIDGE,
SPL0055(
MlOST RECENT VERSION: SEPT 21, 1978.
:
-84-
C
C
SPLIT 2
.:::::::'
:::::::::.::'::::::::::::
C
10
X=A (L+1 ,L+l)
Y=A (LL,
L)
W=A (L,L+1)*.A(L+ 1, L)
P= (Y-X)/2. 0D
Q=P**2+W
IF (Q.GL..O.0D0) GO TO
EI=P+X
E2=DSQRT (-Q)
RETURN
CONTINUE
Z=DSQT (Q)
IF
20
30
40
50
60
70
80
90
(P.LT.0.0ODO)
SPLOO 560
SLOs7
SPL0058C
SPLOO590
SPL0060C
SPL006 10
SPL0062C
SPL00630
SPL0640
SPL0650
SPLO066C
SPL00670
SPL00630
10
SPL0069C
SPL0O700
GO TO 20
Z=P+Z
SPL071C
GO TO 30
CONTINUE
SPL00720
SPLOO73C
Z=P-Z
CONTINUE
IF (Z.EQ.0.010) GO TO 40
R=-W/Z
SPL0074C
SPLO075C
SPL076C
SPL0077C
GO TO 50
SPLOO78C
CONTIN UE
R=O.ODO
CONTINUE
IF (DABS (X+Z).GE.DABS(X+R)) Z=R
Y=Y-X-Z
X=-Z
T=A (L,L+)
U=A (L+ 1,L)
IF (DABS (Y)+DABS(U) .L E.DABS(T)+DABS(X))
Q=U
SPL0079C
SPL080C
SPL008 10
SPL0082C
SPLOO83C
SPLO084C
SPL0085C
SPL0OO86
SPL0087(
SPL00880
GO TO 60
P=Y
SPL0089(
GO TO 70
CONTINUE
Q=X
P=T
CONTINU.E
R=DSQRT (P**2+Q**2)
SPLOO90C
SPL0091 C
SPL0092C
SPL0093(
SPLOO94C
SPL0O95C
IF
SPL0096(
(R.GT.0.ODO)
GO TO 80
E=A (L,L)
E2=A (L+1 ,L+ 1)
A (L+1,L) =0. ODO
RETURN
CONTINUE
P=P/R
Q=Q/R
DO 90 J=L,N
Z=A (L,J)
A (L,J) =PZ +Q*A (L+ 1, J)
A (L+l ,J) =P*A (L-+
t
CONTINUE
Ll=L+1
DO 100 I=1,Lt
,J)-Q*Z
SPL0097C
SPL0098(
SPL0099C
SPL0100(
SPLO101C
SPL0102(
SPLO103(
SPLO104C
SPLO105(
SPLO 106C
SPLO1 07(
SPL0108(
SPLO109(
SPL0110(
-85-
100
110
C
C
C
ZA=A (I,L)
A (I ,L)=P*Z+Q*A (,L+)
A (I, L+ 1)= P*A (I,L+ 1) -Q*Z
CONTINUE
DO 110 i=1,N
Z=V (I,L)
V (I,L)=P*Z+Q*V (I,L+1)
V {I,L+ 1) =P*V (I,L+1) -Q*Z
CONTINUE
A (L+ 1,L)=0.ODO
Et=A (LL)
E2=A (L+t,L+ 1)
RETURN
LAST LINE OF SPLIT
END
SPLIT 3
SPLO 1 1 10
SPL1 120
SPLO 1 130
SPLO1140
SPLO1150
SP01 160
SPLO1 170
SPLO1180
SPLO1190
SPLO0200
SPL01210
SPLO 1220
SPL0123C
SPL01240
SPLO1250
SPLO 1260
SPL01270